research article the explicit identities for spectral...
TRANSCRIPT
Research ArticleThe Explicit Identities for Spectral Norms ofCirculant-Type Matrices Involving BinomialCoefficients and Harmonic Numbers
Jianwei Zhou Xiangyong Chen and Zhaolin Jiang
Department of Mathematics Linyi University Shandong 276005 China
Correspondence should be addressed to Zhaolin Jiang jzh1208sinacom
Received 17 October 2013 Accepted 23 December 2013 Published 12 January 2014
Academic Editor Masoud Hajarian
Copyright copy 2014 Jianwei Zhou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The explicit formulae of spectral norms for circulant-typematrices are investigated thematrices are circulantmatrix skew-circulantmatrix and 119892-circulant matrix respectively The entries are products of binomial coefficients with harmonic numbers Explicitidentities for these spectral norms are obtained Employing these approaches some numerical tests are listed to verify the results
1 Introduction
The classical hypergeometric summation theorems are ex-ploited to derive several striking identities on harmonicnumbers [1] In numerical analysis circulant matrices(named ldquopremultipliersrdquo in numerical methods) are impor-tant because they are diagonalized by a discrete Fouriertransform and hence linear equations that contain themmay be quickly solved using a fast Fourier transform Fur-thermore circulant skew-circulant and 119892-circulant matri-ces play important roles in various applications such asimage processing coding and engineering model For moredetails please refer to [2ndash13] and the references thereinThe skew-circulant matrices were collected to constructpreconditioners for LMF-based ODE codes Hermitian andskew-Hermitian Toeplitz systems were considered in [14ndash17] Lyness employed a skew-circulant matrix to construct 119904-dimensional lattice rules in [18] Recently there are lots ofresearch on the spectral distribution and norms of circulant-type matrices In [19] the authors pointed out the processesbased on the eigenvalue of circulant-type matrices andthe convergence to a Poisson random measure in vaguetopology There were discussions about the convergence inprobability and distribution of the spectral normof circulant-type matrices in [20] The authors in [21] listed the limitingspectral distribution for a class of circulant-type matriceswith heavy tailed input sequence Ngondiep et al showed that
the singular values of 119892-circulants in [22] Solak establishedthe lower and upper bounds for the spectral norms of cir-culant matrices with classical Fibonacci and Lucas numbersentries in [23] Ipek investigated an improved estimation forspectral norms in [24]
In this paper we derive some explicit identities of spectralnorms for some circulant-type matrices with product ofbinomial coefficients with harmonic numbers
The outline of the paper is as follows In Section 2 thedefinitions and preliminary results are listed In Section 3the spectral norms of some circulant matrices are studied InSection 4 the formulae of spectral norms for skew-circulantmatrices are established Section 5 is devoted to investigatethe explicit formulae for 119892-circulant matrices The numericaltests are given in Section 6
2 Preliminaries
The binomial coefficients are defined by (119899
119896) for all natural
numbers 119896 at once by
(1 + 119883)119899
= sum
119896ge0
(119899
119896) 119883119896
(1)
Note that (119899
119896) is the 119896th binomial coefficient of 119899 It is clear
that (119899
0) = 1 (
119899
119899) = 1 and (
119899
119896) = 0 for 119896 gt 119899
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 518913 7 pageshttpdxdoiorg1011552014518913
2 Mathematical Problems in Engineering
The generalized harmonic numbers are defined to bepartial sums of the harmonic series [1]
1198670
(119909) = 0 119867119894(119909) =
119894
sum
119896=0
1
119909 + 119896(119894 = 1 2 ) (2)
For 119909 = 0 in particular they reduce to classical harmonicnumbers
1198670
= 0 119867119894= 1 +
1
2+
1
3+ sdot sdot sdot +
1
119894(119894 = 1 2 )
(3)
We recall the following harmonic number identities [1]119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)
= 2(2119899
119899)
2
(1198672119899
minus 119867119899)
119899
sum
119894=0
(119899
119894) (
2119899
119894) (
3119899 + 119894
119894) (1198673119899+119894
minus 119867119894)
= (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899)
(4)
Definition 1 (see [6 8]) A circulantmatrix is an 119899times119899 complexmatrix with the following form
119860119888
= (
1198860
1198861
sdot sdot sdot 119886119899minus1
119886119899minus1
1198860
sdot sdot sdot 119886119899minus2
119886119899minus2
119886119899minus1
sdot sdot sdot 119886119899minus3
d
1198861
1198862
sdot sdot sdot 1198860
)
119899times119899
(5)
The first row of 119860119888is (1198860 1198861 119886
119899minus1) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by oneposition
Equivalently a circulant matrix can be described withpolynomial as
119860119888
= 119891 (120578119888) =
119899minus1
sum
119894=0
119886119894120578119894
119888 (6)
where
120578119888
= (
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
1 0 0 sdot sdot sdot 0
)
119899times119899
(7)
Obviously 120578119899
119888= 119868119899
Now we discuss the eigenvalues of 119860119888 We declare that
the eigenvalues of 120578119888are the corresponding eigenvalues of 119860
119888
with the function 119891 in (6) which is
120582 (119860119888) = 119891 (120582 (120578
119888)) =
119899minus1
sum
119894=0
119886119894120582(120578119888)119894
(8)
Whereas 120582119895(120578119888) = 120596
119895 (119895 = 0 1 119899 minus 1) then 120582119895(119860119888)
can be calculated by
120582119895(119860119888) =
119899minus1
sum
119894=0
119886119894(120596119895
)119894
(9)
where 120596 = cos(2120587119899) + 119894 sin(2120587119899)Similarly we recall a skew-circulant matrix
Definition 2 (see [6 8]) A skew-circulant matrix is an 119899 times 119899
complex matrix with the following form
119860 sc = (
1198860
1198861
sdot sdot sdot 119886119899minus1
minus119886119899minus1
1198860
sdot sdot sdot 119886119899minus2
minus119886119899minus2
minus119886119899minus1
sdot sdot sdot 119886119899minus3
d
minus1198861
minus1198862
sdot sdot sdot 1198860
)
119899times119899
(10)
Moreover a skew-circulant matrix can be described withpolynomial as
119860 sc = 119891 (120578sc) =
119899minus1
sum
119894=0
119886119894120578119894
sc (11)
where
120578sc = (
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1 0 0 sdot sdot sdot 0
)
119899times119899
(12)
Obviously 120578119899
sc = minus119868119899
Thus we have to calculate the eigenvalues of 119860 sc For thesame reason we obtain that
120582 (119860 sc) = 119891 (120582 (120578sc)) =
119899minus1
sum
119894=0
119886119894120582119894
(120578sc) (13)
Whereas 120582119895(120578sc) = 120596
119895
120572 (119895 = 0 1 119899 minus 1) 120582119895(119860 sc) can
be computed by
120582119895(119860 sc) =
119899minus1
sum
119894=0
119886119894(120596119895
120572)119894
(14)
where 120596 = cos(2120587119899) + 119894 sin(2120587119899) 120572 = cos(120587119899) + 119894 sin(120587119899)
Definition 3 (see [21 25]) A 119892-circulant matrix is an 119899 times 119899
complex matrix with the following form
119860119892
= (
1198860
1198861
119886119899minus1
119886119899minus119892
119886119899minus119892+1
119886119899minus119892minus1
119886119899minus2119892
119886119899minus2119892+1
119886119899minus2119892minus1
d
119886119892
119886119892+1
119886119892minus1
)
119899times119899
(15)
where 119892 is a nonnegative integer and each of the subscripts isunderstood to be reduced modulo 119899
Mathematical Problems in Engineering 3
The first row of 119860119892is (1198860 1198861 119886
119899minus1) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by 119892
positions (equivalently 119892 mod 119899 positions) Note that 119892 = 1
or 119892 = 119899 + 1 yields the standard circulant matrix If 119892 = 119899 minus 1then we obtain the so-called reverse circulant matrix [21]
Definition 4 (see [26]) The spectral norm sdot 2of a matrix
119860 with complex entries is the square root of the largesteigenvalue of the positive semidefinite matrix 119860
lowast
119860
1198602
= radic120582max (119860lowast119860) (16)
where 119860lowast denotes the conjugate transpose of 119860 Therefore if
119860 is an 119899 times 119899 real symmetric matrix or 119860 is a normal matrixthen
1198602
= max1le119894le119899
10038161003816100381610038161205821198941003816100381610038161003816 (17)
where 1205821 1205822 120582
119899are the eigenvalues of 119860
3 Spectral Norms of Some Circulant Matrices
Now we will analyse spectral norms of some given circulantmatrices whose entries are binomial coefficients combinedwith harmonic numbers
Our main results for those matrices are stated as follows
Theorem 5 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198611is as in
(5) and the first row of 1198611is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(18)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then one has
10038171003817100381710038171198611
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (19)
Proof Since circulant matrix 1198611
is normal employingDefinition 4 we claim that the spectral norm of 119861
1is equal to
its spectral radius Furthermore applying the irreducible andentrywise nonnegative properties we claim that 119861
12(ie
its spectral norm) is equal to its Perron value We select an(119899 + 1)-dimensional column vector V = (1 1 1)
119879 then
1198611V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (20)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
1
associated with V which is necessarily the Perron value of 1198611
Employing (4) we obtain
10038171003817100381710038171198611
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (21)
This completes the proof
Hence employing the same approaches we get thefollowing corollary
Corollary 6 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198612be as in
(5) and the first row of 1198612is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(22)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then
10038171003817100381710038171198612
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (23)
Now we investigate some even-order alternative as fol-lows where 119898 is odd (ie 119898 + 1 is even)
Theorem 7 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198613be as in
(5) and the first row of 1198613is
((119898
0)
2
(2119898
0) (1198672119898
minus 1198670)
minus (119898
1)
2
(2119898 + 1
1) (1198672119898+1
minus 1198671)
minus(119898
119898)
2
(2119898 + 119898
119898) (1198672119898+119898
minus 119867119898
))
(24)
where 119886119894
= (minus1)119894
(119898
119894)2
( 2119898+119894119894
) (1198672119898+119894
minus 119867119894) Then there holds
the following identity
10038171003817100381710038171198613
10038171003817100381710038172 = 2(2119898
119898)
2
(1198672119898
minus 119867119898
) (25)
Proof Noticing (9) and (17) it is clear that the spectral normof 1198613can be calculated by
10038171003817100381710038171198613
10038171003817100381710038172 = max0le119905le119898
1003816100381610038161003816120582119905 (1198613)1003816100381610038161003816 = max0le119905le119898
1003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894=0
119886119894(120596119905
)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le max0le119905le119898
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
)119894100381610038161003816100381610038161003816 =
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816
(26)
where 119886119894
= (minus1)119894
(119898
119894)2
( 2119898+119894119894
) (1198672119898+119894
minus 119867119894) and we employed
that all circulant matrices are normalNote that if 119898 is odd then 119898 + 1 is even and 120582
1199050(120578119888) =
1205961199050 = minus1 is an eigenvalue of 120578
119888 so
10038171003817100381710038171198613
10038171003817100381710038172 =
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 (27)
Combining (4) and (27) yields
10038171003817100381710038171198613
10038171003817100381710038172 = 2(2119898
119898)
2
(1198672119898
minus 119867119898
) (28)
This completes the proof
4 Mathematical Problems in Engineering
Employing the same approaches we get the followingcorollary
Corollary 8 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198614be as
in (5) and the first row of 1198614is
((119898
0) (
2119898
0) (
3119898
0) (1198673119898
minus 1198670)
minus (119898
1) (
2119898
1) (
3119898 + 1
1) (1198673119898+1
minus 1198671)
(119898
119898) (
2119898
119898) (
4119898
119898) (1198674119898
minus 119867119898
))
(29)
where 119886119894
= (minus1)119894
(119898
119894) ( 2119898119894
) ( 3119898+119894119894
) (1198673119898+119894
minus 119867119894) Then one has
the following identity
10038171003817100381710038171198614
10038171003817100381710038172 = (3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
) (30)
Similarly we set 1198613
= minus1198613 1198614
= minus1198614
Corollary 9 Let11986131198614be as above respectively and119898 is odd
Then
100381710038171003817100381710038171198613
100381710038171003817100381710038172= 2(
2119898
119898)
2
(1198672119898
minus 119867119898
)
100381710038171003817100381710038171198614
100381710038171003817100381710038172= (
3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
)
(31)
4 Spectral Norms of Skew-Circulant Matrices
An odd-order alternative skew-circulant matrix is defined asfollows where 119904 is even
Theorem 10 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198615be as in
(10) and the first row of 1198615is
((119904
0)
2
(2119904
0) (1198672119904
minus 1198670)
minus(119904
1)
2
(2119904 + 1
1) (1198672119904+1
minus 1198671)
(119904
119904)
2
(2119904 + 119904
119904) (1198672119904+119904
minus 119867119904))
(32)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (33)
Proof We employ (14) and (17) to calculate the spectral normof 1198615as follows for all 119905 = 0 1 119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 =
1003816100381610038161003816100381610038161003816100381610038161003816
119904
sum
119894=0
119886119894(120596119905
120572)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
120572)119894100381610038161003816100381610038161003816
=
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(34)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894)
Since the skew-circulantmatrix is normal we deduce that10038171003817100381710038171198615
10038171003817100381710038172 = max0le119905le119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 (35)
If 119904 is even then 119904 + 1 is odd We declare that 120582sc = minus1
is an eigenvalue of 120578sc then we calculate the correspondingeigenvalue of 119861
5as follows
120582(1198615) =
119904
sum
119894=0
119886119894120582119894
sc =
119904
sum
119894=0
119886119894(minus1)119894
=
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(36)
where we had employed (14)Noticing (34) we claim that 120582
(1198615) is the maximum of
|120582119905(1198615)| which means
10038171003817100381710038171198615
10038171003817100381710038172 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894) (37)
Thus from (4) we obtain
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (38)
This completes the proof
Similarly we can calculate the identity for 1198616
Corollary 11 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198616be as in
(10) and the first row of 1198616is
((119904
0) (
2119904
0) (
3119904
0) (1198673119904
minus 1198670)
minus (119904
1) (
2119904
1) (
3119904 + 1
1) (1198673119904+1
minus 1198671)
(119904
119904) (
2119904
119904) (
3119904 + 119904
119904) (1198673119904+119904
minus 119867119904))
(39)
where 119886119894= (minus1)
119894
(119904
119894) ( 2119904119894
) ( 3119904+119894119894
) (1198673119904+119894
minus 119867119894) Then there holds
10038171003817100381710038171198616
10038171003817100381710038172 = (3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904) (40)
Mathematical Problems in Engineering 5
Corollary 12 Let1198615
= minus1198615and119861
6= minus1198616 and 119904 is evenThen
one has the identities for spectral norm
100381710038171003817100381710038171198615
100381710038171003817100381710038172= 2(
2119904
119904)
2
(21198672119904
minus 119867119904)
100381710038171003817100381710038171198616
100381710038171003817100381710038172= (
3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904)
(41)
5 Spectral Norms of 119892-Circulant Matrices
Inspired by the above propositions we analyse spectral normsof some given 119892-circulant matrices in this section
Lemma 13 (see [25]) The (119899+1)times(119899+1)matrix119876119892is unitary
if and only if
(119899 + 1 119892) = 1 (42)
where 119876119892is a 119892-circulant matrix with the first row 119890
lowast
= [1
0 0]
Lemma 14 (see [25]) 119860 is a 119892-circulant matrix with the firstrow [119886
0 1198861 119886
119899] if and only if
119860 = 119876119892119862 (43)
where
119862 = circ (1198860 1198861 119886
119899) (44)
In the following part we set (119899 + 1 119892) = 1
Theorem 15 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198617be as in
(15) and the first row of 1198617is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(45)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (46)
Proof With the help of Lemmas 13 and 14 we know thatthe 119892-circulant matrix 119861
7is normal then we claim that the
spectral norm of 1198617is equal to its spectral radius Further-
more applying the irreducible and entrywise nonnegativeproperties we claim that 119861
72(ie its spectral norm) is equal
to its Perron value We select a (119899 + 1)-dimensional columnvector V = (1 1 1)
119879 then
1198617V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (47)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
7
associated with V which is necessarily the Perron value of 1198617
Employing (4) we obtain
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (48)
This completes the proof
Corollary 16 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198618be as
in (15) and the first row of 1198618is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(49)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198618
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (50)
6 Numerical Examples
Example 1 In this example we give the numerical results for1198611and 119861
2
Comparing the data in Table 1 we declare that theidentities of spectral norms for 119861
119894(119894 = 1 2) hold
Example 2 In this example we list the numerical results for119861119894 119861119894
(119894 = 3 4)With the help of data in Table 2 it is clear that the
identities of spectral norms for 119861119894 119861119894
(119894 = 3 4) hold
Example 3 In this example we reveal the numerical resultsfor alternative skew-circulant matrices 119861
119894 119861119894
(119894 = 5 6)Combining the data in Table 3 we deduce that the
identities of spectral norms for 119861119894 119861119894
(119894 = 5 6) hold
Example 4 In this example we show numerical results for 1198617
and 1198618
Considering the data in Table 4 we deduce that theidentities of spectral norms for 119861
119894(119894 = 7 8) hold
The above results demonstrate that the identities ofspectral norms for the given matrices hold
7 Conclusion
This paper had discussed the explicit formulae for identicalestimations of spectral norms for circulant skew-circulantand 119892-circulant matrices whose entries are binomial coef-ficients combined with harmonic numbers Furthermoreit is easy to take other entries to obtain more interestingidentities and the same approaches can be used to verifythose identities Furthermore explicit formulas for both
6 Mathematical Problems in Engineering
Table 1 Spectral norms of 119861119894(119894 = 1 2) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 0 1 2 3 4 5 610038171003817100381710038171198611
10038171003817100381710038172 0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 610038171003817100381710038171198612
10038171003817100381710038172 0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
1198621119899
0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 6
1198622119899
0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
Table 2 Spectral norms of 119861119894 119861119894(119894 = 3 4) 119862
1119898= 2(2119898
119898)2
(1198672119898
minus 119867119898
) and 1198622119898
= (3119898
119898)2
(21198673119898
minus 1198672119898
minus 119867119898
)
119898 1 3 5 7 9 1110038171003817100381710038171198613
10038171003817100381710038172 4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 1110038171003817100381710038171198614
10038171003817100381710038172 105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16100381710038171003817100381710038171198613
1003817100381710038171003817100381724 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
100381710038171003817100381710038171198614
100381710038171003817100381710038172105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
1198621119898
4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
1198622119898
105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
Table 3 Spectral norms of 119861119894 119861119894(119894 = 5 6) 119862
1119904= 2(2119904
119904)2
(1198672119904
minus 119867119904) and 119862
2119904= (3119904
119904)2
(21198673119904
minus 1198672119904
minus 119867119904)
119904 0 2 4 6 8 1010038171003817100381710038171198615
10038171003817100381710038172 0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 1010038171003817100381710038171198616
10038171003817100381710038172 0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15100381710038171003817100381710038171198615
1003817100381710038171003817100381720 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
100381710038171003817100381710038171198616
1003817100381710038171003817100381720 296e+2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
1198621119904
0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
1198622119904
0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
Table 4 Spectral norms of 119861119894(119894 = 7 8) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 + 1 5 6 7119892 2 3 4 5 2 3 4 5 610038171003817100381710038171198617
10038171003817100381710038172 622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 610038171003817100381710038171198618
10038171003817100381710038172 344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
1198621119899
622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6
1198622119899
344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
norms 119860 and 119860minus1
help us to estimate the so-calledcondition number It is an interesting problem to investigatethe properties of 119861
119894(119894 = 1 2 8) such as the explicit
formulations for determinants and inverses by just using theentries in the first row
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by NFSC (no 11201212) Promo-tive Research Fund for Excellent Young and Middle-AgedScientists of Shandong Province (no BS2012DX004) Shan-dong Province Higher Educational Science and Technology
Program (J13LI11) AMEP and the Special Funds for DoctoralAuthorities of Linyi University
References
[1] W C Chu and L D Donno ldquoHypergeometric series andharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 34 no 1 pp 123ndash137 2005
[2] M H Ang K T Arasu S Lun Ma and Y Strassler ldquoStudyof proper circulant weighing matrices with weight 9rdquo DiscreteMathematics vol 308 no 13 pp 2802ndash2809 2008
[3] V Brimkov ldquoAlgorithmic and explicit determination of theLovasz number for certain circulant graphsrdquo Discrete AppliedMathematics vol 155 no 14 pp 1812ndash1825 2007
[4] A Cambini ldquoAn explicit form of the inverse of a particularcirculantmatrixrdquoDiscreteMathematics vol 48 no 2-3 pp 323ndash325 1984
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
The generalized harmonic numbers are defined to bepartial sums of the harmonic series [1]
1198670
(119909) = 0 119867119894(119909) =
119894
sum
119896=0
1
119909 + 119896(119894 = 1 2 ) (2)
For 119909 = 0 in particular they reduce to classical harmonicnumbers
1198670
= 0 119867119894= 1 +
1
2+
1
3+ sdot sdot sdot +
1
119894(119894 = 1 2 )
(3)
We recall the following harmonic number identities [1]119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)
= 2(2119899
119899)
2
(1198672119899
minus 119867119899)
119899
sum
119894=0
(119899
119894) (
2119899
119894) (
3119899 + 119894
119894) (1198673119899+119894
minus 119867119894)
= (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899)
(4)
Definition 1 (see [6 8]) A circulantmatrix is an 119899times119899 complexmatrix with the following form
119860119888
= (
1198860
1198861
sdot sdot sdot 119886119899minus1
119886119899minus1
1198860
sdot sdot sdot 119886119899minus2
119886119899minus2
119886119899minus1
sdot sdot sdot 119886119899minus3
d
1198861
1198862
sdot sdot sdot 1198860
)
119899times119899
(5)
The first row of 119860119888is (1198860 1198861 119886
119899minus1) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by oneposition
Equivalently a circulant matrix can be described withpolynomial as
119860119888
= 119891 (120578119888) =
119899minus1
sum
119894=0
119886119894120578119894
119888 (6)
where
120578119888
= (
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
1 0 0 sdot sdot sdot 0
)
119899times119899
(7)
Obviously 120578119899
119888= 119868119899
Now we discuss the eigenvalues of 119860119888 We declare that
the eigenvalues of 120578119888are the corresponding eigenvalues of 119860
119888
with the function 119891 in (6) which is
120582 (119860119888) = 119891 (120582 (120578
119888)) =
119899minus1
sum
119894=0
119886119894120582(120578119888)119894
(8)
Whereas 120582119895(120578119888) = 120596
119895 (119895 = 0 1 119899 minus 1) then 120582119895(119860119888)
can be calculated by
120582119895(119860119888) =
119899minus1
sum
119894=0
119886119894(120596119895
)119894
(9)
where 120596 = cos(2120587119899) + 119894 sin(2120587119899)Similarly we recall a skew-circulant matrix
Definition 2 (see [6 8]) A skew-circulant matrix is an 119899 times 119899
complex matrix with the following form
119860 sc = (
1198860
1198861
sdot sdot sdot 119886119899minus1
minus119886119899minus1
1198860
sdot sdot sdot 119886119899minus2
minus119886119899minus2
minus119886119899minus1
sdot sdot sdot 119886119899minus3
d
minus1198861
minus1198862
sdot sdot sdot 1198860
)
119899times119899
(10)
Moreover a skew-circulant matrix can be described withpolynomial as
119860 sc = 119891 (120578sc) =
119899minus1
sum
119894=0
119886119894120578119894
sc (11)
where
120578sc = (
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1 0 0 sdot sdot sdot 0
)
119899times119899
(12)
Obviously 120578119899
sc = minus119868119899
Thus we have to calculate the eigenvalues of 119860 sc For thesame reason we obtain that
120582 (119860 sc) = 119891 (120582 (120578sc)) =
119899minus1
sum
119894=0
119886119894120582119894
(120578sc) (13)
Whereas 120582119895(120578sc) = 120596
119895
120572 (119895 = 0 1 119899 minus 1) 120582119895(119860 sc) can
be computed by
120582119895(119860 sc) =
119899minus1
sum
119894=0
119886119894(120596119895
120572)119894
(14)
where 120596 = cos(2120587119899) + 119894 sin(2120587119899) 120572 = cos(120587119899) + 119894 sin(120587119899)
Definition 3 (see [21 25]) A 119892-circulant matrix is an 119899 times 119899
complex matrix with the following form
119860119892
= (
1198860
1198861
119886119899minus1
119886119899minus119892
119886119899minus119892+1
119886119899minus119892minus1
119886119899minus2119892
119886119899minus2119892+1
119886119899minus2119892minus1
d
119886119892
119886119892+1
119886119892minus1
)
119899times119899
(15)
where 119892 is a nonnegative integer and each of the subscripts isunderstood to be reduced modulo 119899
Mathematical Problems in Engineering 3
The first row of 119860119892is (1198860 1198861 119886
119899minus1) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by 119892
positions (equivalently 119892 mod 119899 positions) Note that 119892 = 1
or 119892 = 119899 + 1 yields the standard circulant matrix If 119892 = 119899 minus 1then we obtain the so-called reverse circulant matrix [21]
Definition 4 (see [26]) The spectral norm sdot 2of a matrix
119860 with complex entries is the square root of the largesteigenvalue of the positive semidefinite matrix 119860
lowast
119860
1198602
= radic120582max (119860lowast119860) (16)
where 119860lowast denotes the conjugate transpose of 119860 Therefore if
119860 is an 119899 times 119899 real symmetric matrix or 119860 is a normal matrixthen
1198602
= max1le119894le119899
10038161003816100381610038161205821198941003816100381610038161003816 (17)
where 1205821 1205822 120582
119899are the eigenvalues of 119860
3 Spectral Norms of Some Circulant Matrices
Now we will analyse spectral norms of some given circulantmatrices whose entries are binomial coefficients combinedwith harmonic numbers
Our main results for those matrices are stated as follows
Theorem 5 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198611is as in
(5) and the first row of 1198611is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(18)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then one has
10038171003817100381710038171198611
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (19)
Proof Since circulant matrix 1198611
is normal employingDefinition 4 we claim that the spectral norm of 119861
1is equal to
its spectral radius Furthermore applying the irreducible andentrywise nonnegative properties we claim that 119861
12(ie
its spectral norm) is equal to its Perron value We select an(119899 + 1)-dimensional column vector V = (1 1 1)
119879 then
1198611V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (20)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
1
associated with V which is necessarily the Perron value of 1198611
Employing (4) we obtain
10038171003817100381710038171198611
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (21)
This completes the proof
Hence employing the same approaches we get thefollowing corollary
Corollary 6 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198612be as in
(5) and the first row of 1198612is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(22)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then
10038171003817100381710038171198612
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (23)
Now we investigate some even-order alternative as fol-lows where 119898 is odd (ie 119898 + 1 is even)
Theorem 7 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198613be as in
(5) and the first row of 1198613is
((119898
0)
2
(2119898
0) (1198672119898
minus 1198670)
minus (119898
1)
2
(2119898 + 1
1) (1198672119898+1
minus 1198671)
minus(119898
119898)
2
(2119898 + 119898
119898) (1198672119898+119898
minus 119867119898
))
(24)
where 119886119894
= (minus1)119894
(119898
119894)2
( 2119898+119894119894
) (1198672119898+119894
minus 119867119894) Then there holds
the following identity
10038171003817100381710038171198613
10038171003817100381710038172 = 2(2119898
119898)
2
(1198672119898
minus 119867119898
) (25)
Proof Noticing (9) and (17) it is clear that the spectral normof 1198613can be calculated by
10038171003817100381710038171198613
10038171003817100381710038172 = max0le119905le119898
1003816100381610038161003816120582119905 (1198613)1003816100381610038161003816 = max0le119905le119898
1003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894=0
119886119894(120596119905
)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le max0le119905le119898
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
)119894100381610038161003816100381610038161003816 =
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816
(26)
where 119886119894
= (minus1)119894
(119898
119894)2
( 2119898+119894119894
) (1198672119898+119894
minus 119867119894) and we employed
that all circulant matrices are normalNote that if 119898 is odd then 119898 + 1 is even and 120582
1199050(120578119888) =
1205961199050 = minus1 is an eigenvalue of 120578
119888 so
10038171003817100381710038171198613
10038171003817100381710038172 =
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 (27)
Combining (4) and (27) yields
10038171003817100381710038171198613
10038171003817100381710038172 = 2(2119898
119898)
2
(1198672119898
minus 119867119898
) (28)
This completes the proof
4 Mathematical Problems in Engineering
Employing the same approaches we get the followingcorollary
Corollary 8 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198614be as
in (5) and the first row of 1198614is
((119898
0) (
2119898
0) (
3119898
0) (1198673119898
minus 1198670)
minus (119898
1) (
2119898
1) (
3119898 + 1
1) (1198673119898+1
minus 1198671)
(119898
119898) (
2119898
119898) (
4119898
119898) (1198674119898
minus 119867119898
))
(29)
where 119886119894
= (minus1)119894
(119898
119894) ( 2119898119894
) ( 3119898+119894119894
) (1198673119898+119894
minus 119867119894) Then one has
the following identity
10038171003817100381710038171198614
10038171003817100381710038172 = (3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
) (30)
Similarly we set 1198613
= minus1198613 1198614
= minus1198614
Corollary 9 Let11986131198614be as above respectively and119898 is odd
Then
100381710038171003817100381710038171198613
100381710038171003817100381710038172= 2(
2119898
119898)
2
(1198672119898
minus 119867119898
)
100381710038171003817100381710038171198614
100381710038171003817100381710038172= (
3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
)
(31)
4 Spectral Norms of Skew-Circulant Matrices
An odd-order alternative skew-circulant matrix is defined asfollows where 119904 is even
Theorem 10 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198615be as in
(10) and the first row of 1198615is
((119904
0)
2
(2119904
0) (1198672119904
minus 1198670)
minus(119904
1)
2
(2119904 + 1
1) (1198672119904+1
minus 1198671)
(119904
119904)
2
(2119904 + 119904
119904) (1198672119904+119904
minus 119867119904))
(32)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (33)
Proof We employ (14) and (17) to calculate the spectral normof 1198615as follows for all 119905 = 0 1 119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 =
1003816100381610038161003816100381610038161003816100381610038161003816
119904
sum
119894=0
119886119894(120596119905
120572)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
120572)119894100381610038161003816100381610038161003816
=
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(34)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894)
Since the skew-circulantmatrix is normal we deduce that10038171003817100381710038171198615
10038171003817100381710038172 = max0le119905le119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 (35)
If 119904 is even then 119904 + 1 is odd We declare that 120582sc = minus1
is an eigenvalue of 120578sc then we calculate the correspondingeigenvalue of 119861
5as follows
120582(1198615) =
119904
sum
119894=0
119886119894120582119894
sc =
119904
sum
119894=0
119886119894(minus1)119894
=
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(36)
where we had employed (14)Noticing (34) we claim that 120582
(1198615) is the maximum of
|120582119905(1198615)| which means
10038171003817100381710038171198615
10038171003817100381710038172 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894) (37)
Thus from (4) we obtain
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (38)
This completes the proof
Similarly we can calculate the identity for 1198616
Corollary 11 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198616be as in
(10) and the first row of 1198616is
((119904
0) (
2119904
0) (
3119904
0) (1198673119904
minus 1198670)
minus (119904
1) (
2119904
1) (
3119904 + 1
1) (1198673119904+1
minus 1198671)
(119904
119904) (
2119904
119904) (
3119904 + 119904
119904) (1198673119904+119904
minus 119867119904))
(39)
where 119886119894= (minus1)
119894
(119904
119894) ( 2119904119894
) ( 3119904+119894119894
) (1198673119904+119894
minus 119867119894) Then there holds
10038171003817100381710038171198616
10038171003817100381710038172 = (3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904) (40)
Mathematical Problems in Engineering 5
Corollary 12 Let1198615
= minus1198615and119861
6= minus1198616 and 119904 is evenThen
one has the identities for spectral norm
100381710038171003817100381710038171198615
100381710038171003817100381710038172= 2(
2119904
119904)
2
(21198672119904
minus 119867119904)
100381710038171003817100381710038171198616
100381710038171003817100381710038172= (
3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904)
(41)
5 Spectral Norms of 119892-Circulant Matrices
Inspired by the above propositions we analyse spectral normsof some given 119892-circulant matrices in this section
Lemma 13 (see [25]) The (119899+1)times(119899+1)matrix119876119892is unitary
if and only if
(119899 + 1 119892) = 1 (42)
where 119876119892is a 119892-circulant matrix with the first row 119890
lowast
= [1
0 0]
Lemma 14 (see [25]) 119860 is a 119892-circulant matrix with the firstrow [119886
0 1198861 119886
119899] if and only if
119860 = 119876119892119862 (43)
where
119862 = circ (1198860 1198861 119886
119899) (44)
In the following part we set (119899 + 1 119892) = 1
Theorem 15 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198617be as in
(15) and the first row of 1198617is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(45)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (46)
Proof With the help of Lemmas 13 and 14 we know thatthe 119892-circulant matrix 119861
7is normal then we claim that the
spectral norm of 1198617is equal to its spectral radius Further-
more applying the irreducible and entrywise nonnegativeproperties we claim that 119861
72(ie its spectral norm) is equal
to its Perron value We select a (119899 + 1)-dimensional columnvector V = (1 1 1)
119879 then
1198617V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (47)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
7
associated with V which is necessarily the Perron value of 1198617
Employing (4) we obtain
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (48)
This completes the proof
Corollary 16 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198618be as
in (15) and the first row of 1198618is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(49)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198618
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (50)
6 Numerical Examples
Example 1 In this example we give the numerical results for1198611and 119861
2
Comparing the data in Table 1 we declare that theidentities of spectral norms for 119861
119894(119894 = 1 2) hold
Example 2 In this example we list the numerical results for119861119894 119861119894
(119894 = 3 4)With the help of data in Table 2 it is clear that the
identities of spectral norms for 119861119894 119861119894
(119894 = 3 4) hold
Example 3 In this example we reveal the numerical resultsfor alternative skew-circulant matrices 119861
119894 119861119894
(119894 = 5 6)Combining the data in Table 3 we deduce that the
identities of spectral norms for 119861119894 119861119894
(119894 = 5 6) hold
Example 4 In this example we show numerical results for 1198617
and 1198618
Considering the data in Table 4 we deduce that theidentities of spectral norms for 119861
119894(119894 = 7 8) hold
The above results demonstrate that the identities ofspectral norms for the given matrices hold
7 Conclusion
This paper had discussed the explicit formulae for identicalestimations of spectral norms for circulant skew-circulantand 119892-circulant matrices whose entries are binomial coef-ficients combined with harmonic numbers Furthermoreit is easy to take other entries to obtain more interestingidentities and the same approaches can be used to verifythose identities Furthermore explicit formulas for both
6 Mathematical Problems in Engineering
Table 1 Spectral norms of 119861119894(119894 = 1 2) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 0 1 2 3 4 5 610038171003817100381710038171198611
10038171003817100381710038172 0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 610038171003817100381710038171198612
10038171003817100381710038172 0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
1198621119899
0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 6
1198622119899
0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
Table 2 Spectral norms of 119861119894 119861119894(119894 = 3 4) 119862
1119898= 2(2119898
119898)2
(1198672119898
minus 119867119898
) and 1198622119898
= (3119898
119898)2
(21198673119898
minus 1198672119898
minus 119867119898
)
119898 1 3 5 7 9 1110038171003817100381710038171198613
10038171003817100381710038172 4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 1110038171003817100381710038171198614
10038171003817100381710038172 105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16100381710038171003817100381710038171198613
1003817100381710038171003817100381724 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
100381710038171003817100381710038171198614
100381710038171003817100381710038172105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
1198621119898
4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
1198622119898
105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
Table 3 Spectral norms of 119861119894 119861119894(119894 = 5 6) 119862
1119904= 2(2119904
119904)2
(1198672119904
minus 119867119904) and 119862
2119904= (3119904
119904)2
(21198673119904
minus 1198672119904
minus 119867119904)
119904 0 2 4 6 8 1010038171003817100381710038171198615
10038171003817100381710038172 0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 1010038171003817100381710038171198616
10038171003817100381710038172 0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15100381710038171003817100381710038171198615
1003817100381710038171003817100381720 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
100381710038171003817100381710038171198616
1003817100381710038171003817100381720 296e+2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
1198621119904
0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
1198622119904
0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
Table 4 Spectral norms of 119861119894(119894 = 7 8) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 + 1 5 6 7119892 2 3 4 5 2 3 4 5 610038171003817100381710038171198617
10038171003817100381710038172 622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 610038171003817100381710038171198618
10038171003817100381710038172 344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
1198621119899
622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6
1198622119899
344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
norms 119860 and 119860minus1
help us to estimate the so-calledcondition number It is an interesting problem to investigatethe properties of 119861
119894(119894 = 1 2 8) such as the explicit
formulations for determinants and inverses by just using theentries in the first row
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by NFSC (no 11201212) Promo-tive Research Fund for Excellent Young and Middle-AgedScientists of Shandong Province (no BS2012DX004) Shan-dong Province Higher Educational Science and Technology
Program (J13LI11) AMEP and the Special Funds for DoctoralAuthorities of Linyi University
References
[1] W C Chu and L D Donno ldquoHypergeometric series andharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 34 no 1 pp 123ndash137 2005
[2] M H Ang K T Arasu S Lun Ma and Y Strassler ldquoStudyof proper circulant weighing matrices with weight 9rdquo DiscreteMathematics vol 308 no 13 pp 2802ndash2809 2008
[3] V Brimkov ldquoAlgorithmic and explicit determination of theLovasz number for certain circulant graphsrdquo Discrete AppliedMathematics vol 155 no 14 pp 1812ndash1825 2007
[4] A Cambini ldquoAn explicit form of the inverse of a particularcirculantmatrixrdquoDiscreteMathematics vol 48 no 2-3 pp 323ndash325 1984
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
The first row of 119860119892is (1198860 1198861 119886
119899minus1) its (119895 + 1)th row
is obtained by giving its 119895th row a right circular shift by 119892
positions (equivalently 119892 mod 119899 positions) Note that 119892 = 1
or 119892 = 119899 + 1 yields the standard circulant matrix If 119892 = 119899 minus 1then we obtain the so-called reverse circulant matrix [21]
Definition 4 (see [26]) The spectral norm sdot 2of a matrix
119860 with complex entries is the square root of the largesteigenvalue of the positive semidefinite matrix 119860
lowast
119860
1198602
= radic120582max (119860lowast119860) (16)
where 119860lowast denotes the conjugate transpose of 119860 Therefore if
119860 is an 119899 times 119899 real symmetric matrix or 119860 is a normal matrixthen
1198602
= max1le119894le119899
10038161003816100381610038161205821198941003816100381610038161003816 (17)
where 1205821 1205822 120582
119899are the eigenvalues of 119860
3 Spectral Norms of Some Circulant Matrices
Now we will analyse spectral norms of some given circulantmatrices whose entries are binomial coefficients combinedwith harmonic numbers
Our main results for those matrices are stated as follows
Theorem 5 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198611is as in
(5) and the first row of 1198611is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(18)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then one has
10038171003817100381710038171198611
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (19)
Proof Since circulant matrix 1198611
is normal employingDefinition 4 we claim that the spectral norm of 119861
1is equal to
its spectral radius Furthermore applying the irreducible andentrywise nonnegative properties we claim that 119861
12(ie
its spectral norm) is equal to its Perron value We select an(119899 + 1)-dimensional column vector V = (1 1 1)
119879 then
1198611V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (20)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
1
associated with V which is necessarily the Perron value of 1198611
Employing (4) we obtain
10038171003817100381710038171198611
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (21)
This completes the proof
Hence employing the same approaches we get thefollowing corollary
Corollary 6 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198612be as in
(5) and the first row of 1198612is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(22)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then
10038171003817100381710038171198612
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (23)
Now we investigate some even-order alternative as fol-lows where 119898 is odd (ie 119898 + 1 is even)
Theorem 7 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198613be as in
(5) and the first row of 1198613is
((119898
0)
2
(2119898
0) (1198672119898
minus 1198670)
minus (119898
1)
2
(2119898 + 1
1) (1198672119898+1
minus 1198671)
minus(119898
119898)
2
(2119898 + 119898
119898) (1198672119898+119898
minus 119867119898
))
(24)
where 119886119894
= (minus1)119894
(119898
119894)2
( 2119898+119894119894
) (1198672119898+119894
minus 119867119894) Then there holds
the following identity
10038171003817100381710038171198613
10038171003817100381710038172 = 2(2119898
119898)
2
(1198672119898
minus 119867119898
) (25)
Proof Noticing (9) and (17) it is clear that the spectral normof 1198613can be calculated by
10038171003817100381710038171198613
10038171003817100381710038172 = max0le119905le119898
1003816100381610038161003816120582119905 (1198613)1003816100381610038161003816 = max0le119905le119898
1003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894=0
119886119894(120596119905
)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le max0le119905le119898
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
)119894100381610038161003816100381610038161003816 =
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816
(26)
where 119886119894
= (minus1)119894
(119898
119894)2
( 2119898+119894119894
) (1198672119898+119894
minus 119867119894) and we employed
that all circulant matrices are normalNote that if 119898 is odd then 119898 + 1 is even and 120582
1199050(120578119888) =
1205961199050 = minus1 is an eigenvalue of 120578
119888 so
10038171003817100381710038171198613
10038171003817100381710038172 =
119898
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 (27)
Combining (4) and (27) yields
10038171003817100381710038171198613
10038171003817100381710038172 = 2(2119898
119898)
2
(1198672119898
minus 119867119898
) (28)
This completes the proof
4 Mathematical Problems in Engineering
Employing the same approaches we get the followingcorollary
Corollary 8 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198614be as
in (5) and the first row of 1198614is
((119898
0) (
2119898
0) (
3119898
0) (1198673119898
minus 1198670)
minus (119898
1) (
2119898
1) (
3119898 + 1
1) (1198673119898+1
minus 1198671)
(119898
119898) (
2119898
119898) (
4119898
119898) (1198674119898
minus 119867119898
))
(29)
where 119886119894
= (minus1)119894
(119898
119894) ( 2119898119894
) ( 3119898+119894119894
) (1198673119898+119894
minus 119867119894) Then one has
the following identity
10038171003817100381710038171198614
10038171003817100381710038172 = (3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
) (30)
Similarly we set 1198613
= minus1198613 1198614
= minus1198614
Corollary 9 Let11986131198614be as above respectively and119898 is odd
Then
100381710038171003817100381710038171198613
100381710038171003817100381710038172= 2(
2119898
119898)
2
(1198672119898
minus 119867119898
)
100381710038171003817100381710038171198614
100381710038171003817100381710038172= (
3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
)
(31)
4 Spectral Norms of Skew-Circulant Matrices
An odd-order alternative skew-circulant matrix is defined asfollows where 119904 is even
Theorem 10 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198615be as in
(10) and the first row of 1198615is
((119904
0)
2
(2119904
0) (1198672119904
minus 1198670)
minus(119904
1)
2
(2119904 + 1
1) (1198672119904+1
minus 1198671)
(119904
119904)
2
(2119904 + 119904
119904) (1198672119904+119904
minus 119867119904))
(32)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (33)
Proof We employ (14) and (17) to calculate the spectral normof 1198615as follows for all 119905 = 0 1 119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 =
1003816100381610038161003816100381610038161003816100381610038161003816
119904
sum
119894=0
119886119894(120596119905
120572)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
120572)119894100381610038161003816100381610038161003816
=
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(34)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894)
Since the skew-circulantmatrix is normal we deduce that10038171003817100381710038171198615
10038171003817100381710038172 = max0le119905le119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 (35)
If 119904 is even then 119904 + 1 is odd We declare that 120582sc = minus1
is an eigenvalue of 120578sc then we calculate the correspondingeigenvalue of 119861
5as follows
120582(1198615) =
119904
sum
119894=0
119886119894120582119894
sc =
119904
sum
119894=0
119886119894(minus1)119894
=
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(36)
where we had employed (14)Noticing (34) we claim that 120582
(1198615) is the maximum of
|120582119905(1198615)| which means
10038171003817100381710038171198615
10038171003817100381710038172 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894) (37)
Thus from (4) we obtain
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (38)
This completes the proof
Similarly we can calculate the identity for 1198616
Corollary 11 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198616be as in
(10) and the first row of 1198616is
((119904
0) (
2119904
0) (
3119904
0) (1198673119904
minus 1198670)
minus (119904
1) (
2119904
1) (
3119904 + 1
1) (1198673119904+1
minus 1198671)
(119904
119904) (
2119904
119904) (
3119904 + 119904
119904) (1198673119904+119904
minus 119867119904))
(39)
where 119886119894= (minus1)
119894
(119904
119894) ( 2119904119894
) ( 3119904+119894119894
) (1198673119904+119894
minus 119867119894) Then there holds
10038171003817100381710038171198616
10038171003817100381710038172 = (3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904) (40)
Mathematical Problems in Engineering 5
Corollary 12 Let1198615
= minus1198615and119861
6= minus1198616 and 119904 is evenThen
one has the identities for spectral norm
100381710038171003817100381710038171198615
100381710038171003817100381710038172= 2(
2119904
119904)
2
(21198672119904
minus 119867119904)
100381710038171003817100381710038171198616
100381710038171003817100381710038172= (
3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904)
(41)
5 Spectral Norms of 119892-Circulant Matrices
Inspired by the above propositions we analyse spectral normsof some given 119892-circulant matrices in this section
Lemma 13 (see [25]) The (119899+1)times(119899+1)matrix119876119892is unitary
if and only if
(119899 + 1 119892) = 1 (42)
where 119876119892is a 119892-circulant matrix with the first row 119890
lowast
= [1
0 0]
Lemma 14 (see [25]) 119860 is a 119892-circulant matrix with the firstrow [119886
0 1198861 119886
119899] if and only if
119860 = 119876119892119862 (43)
where
119862 = circ (1198860 1198861 119886
119899) (44)
In the following part we set (119899 + 1 119892) = 1
Theorem 15 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198617be as in
(15) and the first row of 1198617is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(45)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (46)
Proof With the help of Lemmas 13 and 14 we know thatthe 119892-circulant matrix 119861
7is normal then we claim that the
spectral norm of 1198617is equal to its spectral radius Further-
more applying the irreducible and entrywise nonnegativeproperties we claim that 119861
72(ie its spectral norm) is equal
to its Perron value We select a (119899 + 1)-dimensional columnvector V = (1 1 1)
119879 then
1198617V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (47)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
7
associated with V which is necessarily the Perron value of 1198617
Employing (4) we obtain
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (48)
This completes the proof
Corollary 16 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198618be as
in (15) and the first row of 1198618is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(49)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198618
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (50)
6 Numerical Examples
Example 1 In this example we give the numerical results for1198611and 119861
2
Comparing the data in Table 1 we declare that theidentities of spectral norms for 119861
119894(119894 = 1 2) hold
Example 2 In this example we list the numerical results for119861119894 119861119894
(119894 = 3 4)With the help of data in Table 2 it is clear that the
identities of spectral norms for 119861119894 119861119894
(119894 = 3 4) hold
Example 3 In this example we reveal the numerical resultsfor alternative skew-circulant matrices 119861
119894 119861119894
(119894 = 5 6)Combining the data in Table 3 we deduce that the
identities of spectral norms for 119861119894 119861119894
(119894 = 5 6) hold
Example 4 In this example we show numerical results for 1198617
and 1198618
Considering the data in Table 4 we deduce that theidentities of spectral norms for 119861
119894(119894 = 7 8) hold
The above results demonstrate that the identities ofspectral norms for the given matrices hold
7 Conclusion
This paper had discussed the explicit formulae for identicalestimations of spectral norms for circulant skew-circulantand 119892-circulant matrices whose entries are binomial coef-ficients combined with harmonic numbers Furthermoreit is easy to take other entries to obtain more interestingidentities and the same approaches can be used to verifythose identities Furthermore explicit formulas for both
6 Mathematical Problems in Engineering
Table 1 Spectral norms of 119861119894(119894 = 1 2) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 0 1 2 3 4 5 610038171003817100381710038171198611
10038171003817100381710038172 0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 610038171003817100381710038171198612
10038171003817100381710038172 0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
1198621119899
0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 6
1198622119899
0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
Table 2 Spectral norms of 119861119894 119861119894(119894 = 3 4) 119862
1119898= 2(2119898
119898)2
(1198672119898
minus 119867119898
) and 1198622119898
= (3119898
119898)2
(21198673119898
minus 1198672119898
minus 119867119898
)
119898 1 3 5 7 9 1110038171003817100381710038171198613
10038171003817100381710038172 4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 1110038171003817100381710038171198614
10038171003817100381710038172 105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16100381710038171003817100381710038171198613
1003817100381710038171003817100381724 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
100381710038171003817100381710038171198614
100381710038171003817100381710038172105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
1198621119898
4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
1198622119898
105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
Table 3 Spectral norms of 119861119894 119861119894(119894 = 5 6) 119862
1119904= 2(2119904
119904)2
(1198672119904
minus 119867119904) and 119862
2119904= (3119904
119904)2
(21198673119904
minus 1198672119904
minus 119867119904)
119904 0 2 4 6 8 1010038171003817100381710038171198615
10038171003817100381710038172 0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 1010038171003817100381710038171198616
10038171003817100381710038172 0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15100381710038171003817100381710038171198615
1003817100381710038171003817100381720 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
100381710038171003817100381710038171198616
1003817100381710038171003817100381720 296e+2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
1198621119904
0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
1198622119904
0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
Table 4 Spectral norms of 119861119894(119894 = 7 8) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 + 1 5 6 7119892 2 3 4 5 2 3 4 5 610038171003817100381710038171198617
10038171003817100381710038172 622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 610038171003817100381710038171198618
10038171003817100381710038172 344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
1198621119899
622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6
1198622119899
344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
norms 119860 and 119860minus1
help us to estimate the so-calledcondition number It is an interesting problem to investigatethe properties of 119861
119894(119894 = 1 2 8) such as the explicit
formulations for determinants and inverses by just using theentries in the first row
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by NFSC (no 11201212) Promo-tive Research Fund for Excellent Young and Middle-AgedScientists of Shandong Province (no BS2012DX004) Shan-dong Province Higher Educational Science and Technology
Program (J13LI11) AMEP and the Special Funds for DoctoralAuthorities of Linyi University
References
[1] W C Chu and L D Donno ldquoHypergeometric series andharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 34 no 1 pp 123ndash137 2005
[2] M H Ang K T Arasu S Lun Ma and Y Strassler ldquoStudyof proper circulant weighing matrices with weight 9rdquo DiscreteMathematics vol 308 no 13 pp 2802ndash2809 2008
[3] V Brimkov ldquoAlgorithmic and explicit determination of theLovasz number for certain circulant graphsrdquo Discrete AppliedMathematics vol 155 no 14 pp 1812ndash1825 2007
[4] A Cambini ldquoAn explicit form of the inverse of a particularcirculantmatrixrdquoDiscreteMathematics vol 48 no 2-3 pp 323ndash325 1984
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Employing the same approaches we get the followingcorollary
Corollary 8 Let (119898 + 1) times (119898 + 1)-circulant matrix 1198614be as
in (5) and the first row of 1198614is
((119898
0) (
2119898
0) (
3119898
0) (1198673119898
minus 1198670)
minus (119898
1) (
2119898
1) (
3119898 + 1
1) (1198673119898+1
minus 1198671)
(119898
119898) (
2119898
119898) (
4119898
119898) (1198674119898
minus 119867119898
))
(29)
where 119886119894
= (minus1)119894
(119898
119894) ( 2119898119894
) ( 3119898+119894119894
) (1198673119898+119894
minus 119867119894) Then one has
the following identity
10038171003817100381710038171198614
10038171003817100381710038172 = (3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
) (30)
Similarly we set 1198613
= minus1198613 1198614
= minus1198614
Corollary 9 Let11986131198614be as above respectively and119898 is odd
Then
100381710038171003817100381710038171198613
100381710038171003817100381710038172= 2(
2119898
119898)
2
(1198672119898
minus 119867119898
)
100381710038171003817100381710038171198614
100381710038171003817100381710038172= (
3119898
119898)
2
(21198673119898
minus 1198672119898
minus 119867119898
)
(31)
4 Spectral Norms of Skew-Circulant Matrices
An odd-order alternative skew-circulant matrix is defined asfollows where 119904 is even
Theorem 10 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198615be as in
(10) and the first row of 1198615is
((119904
0)
2
(2119904
0) (1198672119904
minus 1198670)
minus(119904
1)
2
(2119904 + 1
1) (1198672119904+1
minus 1198671)
(119904
119904)
2
(2119904 + 119904
119904) (1198672119904+119904
minus 119867119904))
(32)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (33)
Proof We employ (14) and (17) to calculate the spectral normof 1198615as follows for all 119905 = 0 1 119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 =
1003816100381610038161003816100381610038161003816100381610038161003816
119904
sum
119894=0
119886119894(120596119905
120572)119894
1003816100381610038161003816100381610038161003816100381610038161003816
le
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 sdot
100381610038161003816100381610038161003816(120596119905
120572)119894100381610038161003816100381610038161003816
=
119904
sum
119894=0
10038161003816100381610038161198861198941003816100381610038161003816 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(34)
where 119886119894= (minus1)
119894
(119904
119894)2
( 2119904+119894119894
) (1198672119904+119894
minus 119867119894)
Since the skew-circulantmatrix is normal we deduce that10038171003817100381710038171198615
10038171003817100381710038172 = max0le119905le119904
1003816100381610038161003816120582119905 (1198615)1003816100381610038161003816 (35)
If 119904 is even then 119904 + 1 is odd We declare that 120582sc = minus1
is an eigenvalue of 120578sc then we calculate the correspondingeigenvalue of 119861
5as follows
120582(1198615) =
119904
sum
119894=0
119886119894120582119894
sc =
119904
sum
119894=0
119886119894(minus1)119894
=
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894)
(36)
where we had employed (14)Noticing (34) we claim that 120582
(1198615) is the maximum of
|120582119905(1198615)| which means
10038171003817100381710038171198615
10038171003817100381710038172 =
119904
sum
119894=0
(119904
119894)
2
(2119904 + 119894
119894) (1198672119904+119894
minus 119867119894) (37)
Thus from (4) we obtain
10038171003817100381710038171198615
10038171003817100381710038172 = 2(2119904
119904)
2
(1198672119904
minus 119867119904) (38)
This completes the proof
Similarly we can calculate the identity for 1198616
Corollary 11 Let (119904 + 1) times (119904 + 1)-circulant matrix 1198616be as in
(10) and the first row of 1198616is
((119904
0) (
2119904
0) (
3119904
0) (1198673119904
minus 1198670)
minus (119904
1) (
2119904
1) (
3119904 + 1
1) (1198673119904+1
minus 1198671)
(119904
119904) (
2119904
119904) (
3119904 + 119904
119904) (1198673119904+119904
minus 119867119904))
(39)
where 119886119894= (minus1)
119894
(119904
119894) ( 2119904119894
) ( 3119904+119894119894
) (1198673119904+119894
minus 119867119894) Then there holds
10038171003817100381710038171198616
10038171003817100381710038172 = (3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904) (40)
Mathematical Problems in Engineering 5
Corollary 12 Let1198615
= minus1198615and119861
6= minus1198616 and 119904 is evenThen
one has the identities for spectral norm
100381710038171003817100381710038171198615
100381710038171003817100381710038172= 2(
2119904
119904)
2
(21198672119904
minus 119867119904)
100381710038171003817100381710038171198616
100381710038171003817100381710038172= (
3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904)
(41)
5 Spectral Norms of 119892-Circulant Matrices
Inspired by the above propositions we analyse spectral normsof some given 119892-circulant matrices in this section
Lemma 13 (see [25]) The (119899+1)times(119899+1)matrix119876119892is unitary
if and only if
(119899 + 1 119892) = 1 (42)
where 119876119892is a 119892-circulant matrix with the first row 119890
lowast
= [1
0 0]
Lemma 14 (see [25]) 119860 is a 119892-circulant matrix with the firstrow [119886
0 1198861 119886
119899] if and only if
119860 = 119876119892119862 (43)
where
119862 = circ (1198860 1198861 119886
119899) (44)
In the following part we set (119899 + 1 119892) = 1
Theorem 15 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198617be as in
(15) and the first row of 1198617is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(45)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (46)
Proof With the help of Lemmas 13 and 14 we know thatthe 119892-circulant matrix 119861
7is normal then we claim that the
spectral norm of 1198617is equal to its spectral radius Further-
more applying the irreducible and entrywise nonnegativeproperties we claim that 119861
72(ie its spectral norm) is equal
to its Perron value We select a (119899 + 1)-dimensional columnvector V = (1 1 1)
119879 then
1198617V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (47)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
7
associated with V which is necessarily the Perron value of 1198617
Employing (4) we obtain
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (48)
This completes the proof
Corollary 16 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198618be as
in (15) and the first row of 1198618is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(49)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198618
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (50)
6 Numerical Examples
Example 1 In this example we give the numerical results for1198611and 119861
2
Comparing the data in Table 1 we declare that theidentities of spectral norms for 119861
119894(119894 = 1 2) hold
Example 2 In this example we list the numerical results for119861119894 119861119894
(119894 = 3 4)With the help of data in Table 2 it is clear that the
identities of spectral norms for 119861119894 119861119894
(119894 = 3 4) hold
Example 3 In this example we reveal the numerical resultsfor alternative skew-circulant matrices 119861
119894 119861119894
(119894 = 5 6)Combining the data in Table 3 we deduce that the
identities of spectral norms for 119861119894 119861119894
(119894 = 5 6) hold
Example 4 In this example we show numerical results for 1198617
and 1198618
Considering the data in Table 4 we deduce that theidentities of spectral norms for 119861
119894(119894 = 7 8) hold
The above results demonstrate that the identities ofspectral norms for the given matrices hold
7 Conclusion
This paper had discussed the explicit formulae for identicalestimations of spectral norms for circulant skew-circulantand 119892-circulant matrices whose entries are binomial coef-ficients combined with harmonic numbers Furthermoreit is easy to take other entries to obtain more interestingidentities and the same approaches can be used to verifythose identities Furthermore explicit formulas for both
6 Mathematical Problems in Engineering
Table 1 Spectral norms of 119861119894(119894 = 1 2) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 0 1 2 3 4 5 610038171003817100381710038171198611
10038171003817100381710038172 0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 610038171003817100381710038171198612
10038171003817100381710038172 0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
1198621119899
0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 6
1198622119899
0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
Table 2 Spectral norms of 119861119894 119861119894(119894 = 3 4) 119862
1119898= 2(2119898
119898)2
(1198672119898
minus 119867119898
) and 1198622119898
= (3119898
119898)2
(21198673119898
minus 1198672119898
minus 119867119898
)
119898 1 3 5 7 9 1110038171003817100381710038171198613
10038171003817100381710038172 4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 1110038171003817100381710038171198614
10038171003817100381710038172 105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16100381710038171003817100381710038171198613
1003817100381710038171003817100381724 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
100381710038171003817100381710038171198614
100381710038171003817100381710038172105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
1198621119898
4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
1198622119898
105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
Table 3 Spectral norms of 119861119894 119861119894(119894 = 5 6) 119862
1119904= 2(2119904
119904)2
(1198672119904
minus 119867119904) and 119862
2119904= (3119904
119904)2
(21198673119904
minus 1198672119904
minus 119867119904)
119904 0 2 4 6 8 1010038171003817100381710038171198615
10038171003817100381710038172 0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 1010038171003817100381710038171198616
10038171003817100381710038172 0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15100381710038171003817100381710038171198615
1003817100381710038171003817100381720 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
100381710038171003817100381710038171198616
1003817100381710038171003817100381720 296e+2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
1198621119904
0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
1198622119904
0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
Table 4 Spectral norms of 119861119894(119894 = 7 8) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 + 1 5 6 7119892 2 3 4 5 2 3 4 5 610038171003817100381710038171198617
10038171003817100381710038172 622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 610038171003817100381710038171198618
10038171003817100381710038172 344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
1198621119899
622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6
1198622119899
344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
norms 119860 and 119860minus1
help us to estimate the so-calledcondition number It is an interesting problem to investigatethe properties of 119861
119894(119894 = 1 2 8) such as the explicit
formulations for determinants and inverses by just using theentries in the first row
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by NFSC (no 11201212) Promo-tive Research Fund for Excellent Young and Middle-AgedScientists of Shandong Province (no BS2012DX004) Shan-dong Province Higher Educational Science and Technology
Program (J13LI11) AMEP and the Special Funds for DoctoralAuthorities of Linyi University
References
[1] W C Chu and L D Donno ldquoHypergeometric series andharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 34 no 1 pp 123ndash137 2005
[2] M H Ang K T Arasu S Lun Ma and Y Strassler ldquoStudyof proper circulant weighing matrices with weight 9rdquo DiscreteMathematics vol 308 no 13 pp 2802ndash2809 2008
[3] V Brimkov ldquoAlgorithmic and explicit determination of theLovasz number for certain circulant graphsrdquo Discrete AppliedMathematics vol 155 no 14 pp 1812ndash1825 2007
[4] A Cambini ldquoAn explicit form of the inverse of a particularcirculantmatrixrdquoDiscreteMathematics vol 48 no 2-3 pp 323ndash325 1984
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Corollary 12 Let1198615
= minus1198615and119861
6= minus1198616 and 119904 is evenThen
one has the identities for spectral norm
100381710038171003817100381710038171198615
100381710038171003817100381710038172= 2(
2119904
119904)
2
(21198672119904
minus 119867119904)
100381710038171003817100381710038171198616
100381710038171003817100381710038172= (
3119904
119904)
2
(21198673119904
minus 1198672119904
minus 119867119904)
(41)
5 Spectral Norms of 119892-Circulant Matrices
Inspired by the above propositions we analyse spectral normsof some given 119892-circulant matrices in this section
Lemma 13 (see [25]) The (119899+1)times(119899+1)matrix119876119892is unitary
if and only if
(119899 + 1 119892) = 1 (42)
where 119876119892is a 119892-circulant matrix with the first row 119890
lowast
= [1
0 0]
Lemma 14 (see [25]) 119860 is a 119892-circulant matrix with the firstrow [119886
0 1198861 119886
119899] if and only if
119860 = 119876119892119862 (43)
where
119862 = circ (1198860 1198861 119886
119899) (44)
In the following part we set (119899 + 1 119892) = 1
Theorem 15 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198617be as in
(15) and the first row of 1198617is
((119899
0)
2
(2119899
0) (1198672119899
minus 1198670)
(119899
1)
2
(2119899 + 1
1) (1198672119899+1
minus 1198671)
(119899
119899)
2
(2119899 + 119899
119899) (1198672119899+119899
minus 119867119899))
(45)
where 119886119894= (119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus 119867119894) Then
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (46)
Proof With the help of Lemmas 13 and 14 we know thatthe 119892-circulant matrix 119861
7is normal then we claim that the
spectral norm of 1198617is equal to its spectral radius Further-
more applying the irreducible and entrywise nonnegativeproperties we claim that 119861
72(ie its spectral norm) is equal
to its Perron value We select a (119899 + 1)-dimensional columnvector V = (1 1 1)
119879 then
1198617V = (
119899
sum
119894=0
(119899
119894)
2
(2119899 + 119894
119894) (1198672119899+119894
minus 119867119894)) V (47)
Obviouslysum119899119894=0
(119899
119894)2
( 2119899+119894119894
) (1198672119899+119894
minus119867119894) is an eigenvalue of 119861
7
associated with V which is necessarily the Perron value of 1198617
Employing (4) we obtain
10038171003817100381710038171198617
10038171003817100381710038172 = 2(2119899
119899)
2
(1198672119899
minus 119867119899) (48)
This completes the proof
Corollary 16 Let (119899 + 1) times (119899 + 1)-circulant matrix 1198618be as
in (15) and the first row of 1198618is
((119899
0) (
2119899
0) (
3119899
0) (1198673119899
minus 1198670)
(119899
1) (
2119899
1) (
3119899 + 1
1) (1198673119899+1
minus 1198671)
(119899
119899) (
2119899
119899) (
4119899
119899) (1198674119899
minus 119867119899))
(49)
where 119886119894= (119899
119894) ( 2119899119894
) ( 3119899+119894119894
) (1198673119899+119894
minus 119867119894) Then one obtains
10038171003817100381710038171198618
10038171003817100381710038172 = (3119899
119899)
2
(21198673119899
minus 1198672119899
minus 119867119899) (50)
6 Numerical Examples
Example 1 In this example we give the numerical results for1198611and 119861
2
Comparing the data in Table 1 we declare that theidentities of spectral norms for 119861
119894(119894 = 1 2) hold
Example 2 In this example we list the numerical results for119861119894 119861119894
(119894 = 3 4)With the help of data in Table 2 it is clear that the
identities of spectral norms for 119861119894 119861119894
(119894 = 3 4) hold
Example 3 In this example we reveal the numerical resultsfor alternative skew-circulant matrices 119861
119894 119861119894
(119894 = 5 6)Combining the data in Table 3 we deduce that the
identities of spectral norms for 119861119894 119861119894
(119894 = 5 6) hold
Example 4 In this example we show numerical results for 1198617
and 1198618
Considering the data in Table 4 we deduce that theidentities of spectral norms for 119861
119894(119894 = 7 8) hold
The above results demonstrate that the identities ofspectral norms for the given matrices hold
7 Conclusion
This paper had discussed the explicit formulae for identicalestimations of spectral norms for circulant skew-circulantand 119892-circulant matrices whose entries are binomial coef-ficients combined with harmonic numbers Furthermoreit is easy to take other entries to obtain more interestingidentities and the same approaches can be used to verifythose identities Furthermore explicit formulas for both
6 Mathematical Problems in Engineering
Table 1 Spectral norms of 119861119894(119894 = 1 2) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 0 1 2 3 4 5 610038171003817100381710038171198611
10038171003817100381710038172 0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 610038171003817100381710038171198612
10038171003817100381710038172 0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
1198621119899
0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 6
1198622119899
0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
Table 2 Spectral norms of 119861119894 119861119894(119894 = 3 4) 119862
1119898= 2(2119898
119898)2
(1198672119898
minus 119867119898
) and 1198622119898
= (3119898
119898)2
(21198673119898
minus 1198672119898
minus 119867119898
)
119898 1 3 5 7 9 1110038171003817100381710038171198613
10038171003817100381710038172 4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 1110038171003817100381710038171198614
10038171003817100381710038172 105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16100381710038171003817100381710038171198613
1003817100381710038171003817100381724 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
100381710038171003817100381710038171198614
100381710038171003817100381710038172105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
1198621119898
4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
1198622119898
105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
Table 3 Spectral norms of 119861119894 119861119894(119894 = 5 6) 119862
1119904= 2(2119904
119904)2
(1198672119904
minus 119867119904) and 119862
2119904= (3119904
119904)2
(21198673119904
minus 1198672119904
minus 119867119904)
119904 0 2 4 6 8 1010038171003817100381710038171198615
10038171003817100381710038172 0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 1010038171003817100381710038171198616
10038171003817100381710038172 0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15100381710038171003817100381710038171198615
1003817100381710038171003817100381720 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
100381710038171003817100381710038171198616
1003817100381710038171003817100381720 296e+2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
1198621119904
0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
1198622119904
0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
Table 4 Spectral norms of 119861119894(119894 = 7 8) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 + 1 5 6 7119892 2 3 4 5 2 3 4 5 610038171003817100381710038171198617
10038171003817100381710038172 622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 610038171003817100381710038171198618
10038171003817100381710038172 344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
1198621119899
622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6
1198622119899
344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
norms 119860 and 119860minus1
help us to estimate the so-calledcondition number It is an interesting problem to investigatethe properties of 119861
119894(119894 = 1 2 8) such as the explicit
formulations for determinants and inverses by just using theentries in the first row
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by NFSC (no 11201212) Promo-tive Research Fund for Excellent Young and Middle-AgedScientists of Shandong Province (no BS2012DX004) Shan-dong Province Higher Educational Science and Technology
Program (J13LI11) AMEP and the Special Funds for DoctoralAuthorities of Linyi University
References
[1] W C Chu and L D Donno ldquoHypergeometric series andharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 34 no 1 pp 123ndash137 2005
[2] M H Ang K T Arasu S Lun Ma and Y Strassler ldquoStudyof proper circulant weighing matrices with weight 9rdquo DiscreteMathematics vol 308 no 13 pp 2802ndash2809 2008
[3] V Brimkov ldquoAlgorithmic and explicit determination of theLovasz number for certain circulant graphsrdquo Discrete AppliedMathematics vol 155 no 14 pp 1812ndash1825 2007
[4] A Cambini ldquoAn explicit form of the inverse of a particularcirculantmatrixrdquoDiscreteMathematics vol 48 no 2-3 pp 323ndash325 1984
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 1 Spectral norms of 119861119894(119894 = 1 2) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 0 1 2 3 4 5 610038171003817100381710038171198611
10038171003817100381710038172 0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 610038171003817100381710038171198612
10038171003817100381710038172 0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
1198621119899
0 4 42 493119890 + 2 622119890 + 3 820119890 + 4 112119890 + 6
1198622119899
0 105119890 + 1 296119890 + 2 969119890 + 3 344119890 + 5 128119890 + 7 495119890 + 8
Table 2 Spectral norms of 119861119894 119861119894(119894 = 3 4) 119862
1119898= 2(2119898
119898)2
(1198672119898
minus 119867119898
) and 1198622119898
= (3119898
119898)2
(21198673119898
minus 1198672119898
minus 119867119898
)
119898 1 3 5 7 9 1110038171003817100381710038171198613
10038171003817100381710038172 4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 1110038171003817100381710038171198614
10038171003817100381710038172 105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16100381710038171003817100381710038171198613
1003817100381710038171003817100381724 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
100381710038171003817100381710038171198614
100381710038171003817100381710038172105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
1198621119898
4 493119890 + 2 820119890 + 4 155119890 + 7 315119890 + 9 668119890 + 11
1198622119898
105 969119890 + 3 128119890 + 7 196119890 + 10 320119890 + 13 549119890 + 16
Table 3 Spectral norms of 119861119894 119861119894(119894 = 5 6) 119862
1119904= 2(2119904
119904)2
(1198672119904
minus 119867119904) and 119862
2119904= (3119904
119904)2
(21198673119904
minus 1198672119904
minus 119867119904)
119904 0 2 4 6 8 1010038171003817100381710038171198615
10038171003817100381710038172 0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 1010038171003817100381710038171198616
10038171003817100381710038172 0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15100381710038171003817100381710038171198615
1003817100381710038171003817100381720 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
100381710038171003817100381710038171198616
1003817100381710038171003817100381720 296e+2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
1198621119904
0 42 622119890 + 3 112119890 + 6 220119890 + 8 457119890 + 10
1198622119904
0 296119890 + 2 344119890 + 5 495119890 + 5 786119890 + 11 132119890 + 15
Table 4 Spectral norms of 119861119894(119894 = 7 8) 119862
1119899= 2(2119899
119899)2
(1198672119899
minus 119867119899) and 119862
2119899= (3119899
119899)2
(21198673119899
minus 1198672119899
minus 119867119899)
119899 + 1 5 6 7119892 2 3 4 5 2 3 4 5 610038171003817100381710038171198617
10038171003817100381710038172 622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 610038171003817100381710038171198618
10038171003817100381710038172 344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
1198621119899
622119890 + 3 622119890 + 3 622119890 + 3 820119890 + 4 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6 112119890 + 6
1198622119899
344119890 + 5 344119890 + 5 344119890 + 5 128119890 + 7 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8 495119890 + 8
norms 119860 and 119860minus1
help us to estimate the so-calledcondition number It is an interesting problem to investigatethe properties of 119861
119894(119894 = 1 2 8) such as the explicit
formulations for determinants and inverses by just using theentries in the first row
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by NFSC (no 11201212) Promo-tive Research Fund for Excellent Young and Middle-AgedScientists of Shandong Province (no BS2012DX004) Shan-dong Province Higher Educational Science and Technology
Program (J13LI11) AMEP and the Special Funds for DoctoralAuthorities of Linyi University
References
[1] W C Chu and L D Donno ldquoHypergeometric series andharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 34 no 1 pp 123ndash137 2005
[2] M H Ang K T Arasu S Lun Ma and Y Strassler ldquoStudyof proper circulant weighing matrices with weight 9rdquo DiscreteMathematics vol 308 no 13 pp 2802ndash2809 2008
[3] V Brimkov ldquoAlgorithmic and explicit determination of theLovasz number for certain circulant graphsrdquo Discrete AppliedMathematics vol 155 no 14 pp 1812ndash1825 2007
[4] A Cambini ldquoAn explicit form of the inverse of a particularcirculantmatrixrdquoDiscreteMathematics vol 48 no 2-3 pp 323ndash325 1984
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
[5] W S Chou B S Du and P J S Shiue ldquoA note on circulanttransition matrices in Markov chainsrdquo Linear Algebra and ItsApplications vol 429 no 7 pp 1699ndash1704 2008
[6] P Davis Circulant Matrices Wiley New York NY USA 1979[7] C Erbas and M M Tanik ldquoGenerating solutions to the N-
Queens problems using 2-circulantsrdquo Mathematics Magazinevol 68 no 5 pp 343ndash356 1995
[8] Z L Jiang and Z X Zhou Circulant Matrices ChengduUniversity of Science and Technology Press Chengdu China1999
[9] A Mamut Q X Huang and F J Liu ldquoEnumeration of 2-regular circulant graphs and directed double networksrdquoDiscreteApplied Mathematics vol 157 no 5 pp 1024ndash1033 2009
[10] I Stojmenovic ldquoMultiplicative circulant networks topologicalproperties and communication algorithmsrdquo Discrete AppliedMathematics vol 77 no 3 pp 281ndash305 1997
[11] M Ventou and C Rigoni ldquoSelf-dual doubly circulant codesrdquoDiscrete Mathematics vol 56 no 2-3 pp 291ndash298 1985
[12] M J Weinberger and A Lempel ldquoFactorization of symmetriccirculantmatrices in finite fieldsrdquoDiscrete AppliedMathematicsvol 28 no 3 pp 271ndash285 1990
[13] Y K Wu R Z Jia and Q Li ldquog-circulant solutions to the (0 1)matrix equation 119860
119898
= 119869lowast
119899rdquo Linear Algebra and its Applications
vol 345 no 1ndash3 pp 195ndash224 2002[14] D Bertaccini and M K Ng ldquoSkew-circulant preconditioners
for systems of LMF-based ODE codesrdquo in Numerical Analysisand Its Applications Lecture Notes in Computer Science pp93ndash101 2001
[15] R Chan and X Q Jin ldquoCirculant and skew-circulant pre-conditioners for skew-hermitian type Toeplitz systemsrdquo BITNumerical Mathematics vol 31 no 4 pp 632ndash646 1991
[16] R Chan andM K Ng ldquoToeplitz preconditioners for HermitianToeplitz systemsrdquo Linear Algebra and Its Applications C vol 190pp 181ndash208 1993
[17] T Huclke ldquoCirculant and skew-circulant matrices for solvingToeplitz matrix problemsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 767ndash777 1992
[18] J N Lyness and T SOslashrevik ldquoFour-dimensional lattice rulesgenerated by skew-circulant matricesrdquoMathematics of Compu-tation vol 73 no 245 pp 279ndash295 2004
[19] A Bose R S Hazra and K Saha ldquoPoisson convergence ofeigenvalues of circulant type matricesrdquo Extremes vol 14 no 4pp 365ndash392 2011
[20] A Bose R S Hazra and K Saha ldquoSpectral norm of circulant-type matricesrdquo Journal of Theoretical Probability vol 24 no 2pp 479ndash516 2011
[21] A Bose S Guha R S Hazra and K Saha ldquoCirculant typematrices with heavy tailed entriesrdquo Statistics and ProbabilityLetters vol 81 no 11 pp 1706ndash1716 2011
[22] E Ngondiep S Serra-Capizzano and D Sesana ldquoSpectralfeatures and asymptotic properties for g-circulants and g-Toeplitz sequencesrdquo SIAM Journal on Matrix Analysis andApplications vol 31 no 4 pp 1663ndash1687 2009
[23] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005
[24] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011
[25] W T Stallings and T L Boullion ldquoThe pseudoinverse of anr-circulant matrixrdquo Proceedings of the American MathematicalSociety vol 34 no 2 pp 385ndash388 1972
[26] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1985
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of