research article isomorphic operators and functional...

Research ArticleIsomorphic Operators and Functional Equations forthe Skew-Circulant Algebra

Zhaolin Jiang,1 Tingting Xu,1,2 and Fuliang Lu1

1 Department of Mathematics, Linyi University, Linyi, Shandong 276000, China2Department of Mathematics, Shandong Normal University, Ji’nan, Shandong 250014, China

Correspondence should be addressed to Tingting Xu; [email protected]

Received 28 March 2014; Accepted 27 April 2014; Published 8 May 2014

Academic Editor: Tongxing Li

Copyright © 2014 Zhaolin Jiang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants withcomplex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on theskew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of 𝑛 × 𝑛

complex skew-circulant matrices are displayed in this paper.

1. Introduction

Skew circulant and circulantmatrices have became importanttools in solving various differential equations. Bertaccini andNg [1] proposed a nonsingular skew-circulant preconditionerfor systems of LMF-based ODE codes. Delgado et al. [2]developed some techniques to obtain global hyperbolicity fora certain class of endomorphisms of (𝑅𝑝)𝑛 with 𝑝, 𝑛 ≥ 2;this kind of endomorphisms is obtained from vectorial differ-ence equations where the mapping defining these equationssatisfies a circulant matrix condition. Wilde [3] developeda theory for the solution of ordinary and partial differentialequations whose structure involves the algebra of circulants.He showed how the algebra of 2 × 2 circulants relates tothe study of the harmonic oscillator, the Cauchy-Riemannequations, Laplace’s equation, the Lorentz transformation,and the wave equation. And he used 𝑛 × 𝑛 circulants tosuggest natural generalizations of these equations to higherdimensions. Using circulant matrix, Karasozen and Simsek[4] considered periodic boundary conditions such that noadditional boundary terms will appear after semidiscretiza-tion. In [5], the resulting dense linear system exhibits somuchstructure that it can be solved very efficiently by a circulantpreconditioned conjugate gradient method. Brockett andWillems [6] showed how the important problems of linearsystem theory can be solved concisely for a particular class of

linear systems, namely, block circulant systems, by exploitingthe algebraic structure. Circulant matrices were also used tosolve linear systems from differential-algebraic equations anddelay differential equations; see [7, 8].

Skew circulant matrices have important applications invarious disciplines including image processing, communica-tions, signal processing, encoding, solving Toeplitz matrixproblems, preconditioner, and solving least squares problems.They have been put on firm basis with the work of Davis[9] and Jiang and Zhou [10]. Hermitian and skew-HermitianToeplitz systems are considered in [11–13]. Lyness and Sørevik[14] employed a skew circulant matrix to construct 𝑠-dimensional lattice rules. Spectral decompositions of skewcirculant and skew left circulant matrices were discussed in[15]. Compared with cyclic convolution algorithm, the skewcyclic convolution algorithm [16] is able to perform filteringprocedure in approximately half of computational cost forreal signals. In [17] two new normal-form realizations arepresented by utilizing circulant and skew circulant matricesas their state transition matrices. The well-known second-order coupled form is a special case of the skew circulantform. Li et al. [18] gave the style spectral decompositionof skew circulant matrix firstly and then dealt with theoptimal backward perturbation analysis for the linear systemwith skew circulant coefficient matrix. In [19], a new fast

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 418194, 8 pageshttp://dx.doi.org/10.1155/2014/418194

2 Abstract and Applied Analysis

algorithm for optimal design of block digital filters (BDFs)is proposed based on skew circulant matrix. Gao et al. [20]gave explicit determinants and inverses of skew circulantand skew left circulant matrices with Fibonacci and Lucasnumbers.

Besides, there are several papers on the circulantoperator and circulant algebra. Wilde [21] discussedaspects of functional equations obtaining generalizations ofodd and even functions in terms of 𝑛th roots of unity incomplex number field. Wilde [22, 23] generalized propertiesof 2 × 2 circulant matrices and 2-dimensional complexanalysis to 𝑛 × 𝑛 circulant matrices. Wilde [24] displayedalgebras of operators which are isomorphic to the algebraof 𝑛 × 𝑛 complex circulant matrices. Chan et al. [25] gavedifferent formulations of the operator, discussed its algebraicand geometric properties, and computed its operatornorms in different Banach algebras of matrices. Using theseresults, they also gave an efficient algorithm for finding thesuperoptimal circulant preconditioner. Chillag [26] provedthat eigenvalues of some generalized circulant matricesand the Matteson-Solomon coefficients of a codeword ofa cyclic code are all examples of eigenvalues of elementsof semisimple finite-dimensional, commutative algebras.The purpose of Chillag’s paper is to exhibit elementaryproperties of such algebras and to apply these properties invarious situations. Brink and Pretorius [27] embed someresults on Boolean circulants into the context of relationalgebras and then generalised them. Interestingly enough,the route lies in group theory. Chan et al. [28] studied thesolutions of finite-section Wiener-Hopf equations, by thepreconditioned conjugate gradient method, and gave aneasy and general scheme of constructing good circulantintegral operators as preconditioners for such equations.Benedetto and Capizzano [29] investigated algebraic andgeometric properties of the optimal approximation operator,generalizing those proved in [25] for the basic circulant case.Benedetto and Capizzano [30] considered the superoptimalFrobenius operators in several matrix vector spaces and inparticular in the circulant algebra, by emphasizing both thealgebraic and geometric properties. Hwang et al. [31] areconcerned with the hyponormality of Toeplitz operatorswith matrix-valued circulant symbols. They established anecessary and sufficient condition for Toeplitz operators withmatrix-valued partially circulant symbols to be hyponormaland provided a rank formula for the self-commutator. Severalnorm equalities and inequalities for operator matrices areproved in [32]. These results, which depend on the structureof circulant and skew circulant operator matrices, includepinching type inequalities for weakly unitarily invariantnorms.

In passing, skew-circulant operator and algebra were onlyused in [32]. And solving differential equations by skewcirculant matrices has not been fully exploited (as far as weknown, only in [1]). It is hoped that this paper will help inchanging this. More work continuing the present paper isforthcoming.

In Section 2, we will prove that a complex 𝑛 × 𝑛 skew-circulant matrix is a matrix representation of the group ring(over C) of the skew cyclic group. We also prove that the set

of skew-circulants with complex entries has an idempotentbasis.

In Section 3, functional equations, whose solutions arefunctions C𝑛 → C, are solved using skew-cyclic andidempotent linear operators on the space (labeled 𝑈) offunctions C𝑛 → C. Further, we will show that this algebraof linear operators is isomorphic to 𝑛 × 𝑛 skew-circulants.

In Section 4, we display skew cyclic and idempotent linearoperators on the space𝑉 of functions on 𝑛×𝑛 complex skew-circulants. Furthermore, we get a relationship between theoperators on 𝑉 and those on 𝑈.

In Section 5, we show a linear involution on 𝑉 whosegroup ring is isomorphic to 2 × 2 complex skew-circulantmatrices.

2. Properties of Skew-Circulant

An 𝑛 × 𝑛 skew-circulant matrix is a square matrix like thefollowing:

𝑆 =(

(

𝑥0 𝑥1 . . . 𝑥𝑛−1

−𝑥𝑛−1 𝑥0 . . . 𝑥𝑛−2

−𝑥𝑛−2 −𝑥𝑛−1 ⋅ ⋅ ⋅ 𝑥𝑛−3

......

......

−𝑥1 −𝑥2 ⋅ ⋅ ⋅ 𝑥0

)

)𝑛×𝑛

. (1)

Let 𝑆𝑛 denote the set of skew-circulant matrices withcomplex entries. Let 𝜋 denote the skew-circulant matrix with𝑥1 = 1 and 𝑥𝑗 = 0 for 𝑗 = 1. Then 𝜋

𝑙 (the 𝑙th power of 𝜋,1 ≤ 𝑙 < 𝑛) is the skew-circulant matrix with 𝑥𝑙 = 1 and 𝑥𝑗 = 0

for all 𝑗 = 𝑙, 𝜋0 = 𝐼 (𝐼 is the identity matrix), 𝜋1 = 𝜋, and𝜋𝑛= −𝐼. So 𝑆 can be written as

𝑆 =

𝑛−1

∑

𝑙=0

𝑥𝑙𝜋𝑙, (2)

for 𝑥0, 𝑥1, . . . , 𝑥𝑛−1 ∈ C. In other words, 𝜋0, 𝜋1, 𝜋2, . . . , 𝜋𝑛−1form a basis for the set of skew-circulant matrices. Let 𝜔

denote any of the 𝑛th roots of unit, or 𝜔 = 𝑒2𝜋𝑖/𝑛, 𝜂 = 𝑒

𝜋𝑖/𝑛,and

𝑦𝑙 =

𝑛−1

∑

𝑗=0

𝜂𝑗𝜔𝑙𝑗𝑥𝑗, for 𝑙 = 0, 1, . . . , 𝑛 − 1. (3)

Then, through the eigenvalues of the matrix 𝜋𝑖, 𝑖 =

0, 1, . . . , 𝑛 − 1, we know that the numbers 𝑦0, 𝑦1, . . . , 𝑦𝑛−1 arethe eigenvalues of the skew-circulant matrix 𝑆.

Proposition 1. If 𝑆𝑛, the set of skew-circulant matrices, hasa basis 𝜋0, 𝜋1, 𝜋2, . . . , 𝜋𝑛−1, then the 𝑆𝑛 also has another basis𝐺0, 𝐺1, . . . , 𝐺𝑛−1, where

𝐺𝑙 =1

𝑛

𝑛−1

∑

𝑗=0

(𝜂−𝑗𝜔−𝑙𝑗

𝜋𝑗) , (4)

for 𝜔 = 𝑒2𝜋𝑖/𝑛, 𝜂 = 𝑒

𝜋𝑖/𝑛, and 𝑙 = 0, 1, . . . , 𝑛 − 1.

Abstract and Applied Analysis 3

Proof. By calculation, these matrices𝐺𝑙 (𝑙 = 0, 1, 2, . . . , 𝑛 − 1)

have the following properties:

𝐺2

𝑙= 𝐺𝑙, for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1;

𝐺𝑙𝐺𝑖 = 0, for 𝑙 = 𝑖;

𝐺0 + 𝐺1 + 𝐺2 + ⋅ ⋅ ⋅ + 𝐺𝑛−1 = 𝐼;

𝜋𝑙=

𝑛−1

∑

𝑗=0

𝜂𝑗𝜔𝑙𝑗𝐺𝑗,

(5)

for 𝑙 = 0, 1, . . . , 𝑛 − 1. Thus, the idempotents 𝐺0, 𝐺1, . . . , 𝐺𝑛−1form a basis for 𝑆𝑛, and in (2) we also have

𝑆 =

𝑛−1

∑

𝑙=0

𝑦𝑙𝐺𝑙 = 𝑦0𝐺0 + ⋅ ⋅ ⋅ + 𝑦𝑛−1𝐺𝑛−1. (6)

We have seen that every skew-circulant matrix, 𝑆 ∈ 𝑆𝑛,can be written in one and only one way in the form (2); thatis

𝑆 = 𝑥0𝐼 + 𝑥1𝜋 + 𝑥2𝜋2+ ⋅ ⋅ ⋅ + 𝑥𝑛−1𝜋

𝑛−1. (7)

Proposition 2. If the function 𝜙 : 𝑆𝑛 → 𝑆𝑛 is defined by

𝜙 (𝑆) = 𝜂𝑥0𝐼 + 𝜂𝜔𝑥1𝜋 + ⋅ ⋅ ⋅ + 𝜂𝜔𝑛−1

𝑥𝑛−1𝜋𝑛−1

, (8)

then 𝜙𝑛(𝑆) = −𝑆.

Proof. By composition, we gain that

𝜙𝑘(𝑆) = 𝜂

𝑘𝑥0𝐼 + 𝜂

𝑘𝜔𝑘𝑥1𝜋 + 𝜂

𝑘𝜔2𝑘𝑥2𝜋2

+ ⋅ ⋅ ⋅ + 𝜂𝑘𝜔(𝑛−1)𝑘


;

(9)

that is, 𝜙𝑘 replaces 𝑥𝑙 by 𝜂𝑘𝜔𝑙𝑘𝑥𝑙 for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1. Also,

𝜙𝑛(𝑆) = −𝑆.

The function𝜙 is an automorphism in 𝑆𝑛 that preservesC,with C being embedded in 𝑆𝑛 by the correspondence 𝑐 → 𝑐𝐼

for 𝑐 ∈ C.

Proposition 3. Let 𝑝𝑙 be the function 𝑆𝑛 → 𝑆𝑛 defined by

𝑝𝑙 =1

𝑛

𝑛−1

∑

𝑗=0

𝜂−𝑗𝜔−𝑙𝑗

𝜙𝑗, 𝑓𝑜𝑟 𝑙 = 0, 1, . . . , 𝑛 − 1. (10)

Then,

𝑝2

𝑙= 𝑝𝑙, 𝑓𝑜𝑟 𝑙 = 0, 1, 2, . . . , 𝑛 − 1;

𝑝𝑙𝑝𝑗 = 0, 𝑓𝑜𝑟 𝑙 = 𝑗;

𝑝0 + 𝑝1 + 𝑝2 + ⋅ ⋅ ⋅ + 𝑝𝑛−1 = 𝜙0;

𝜂𝑙𝑝0 + 𝜂

𝑙𝜔𝑙𝑝1 + ⋅ ⋅ ⋅ + 𝜂

𝑙𝜔(𝑛−1)𝑙

𝑝𝑛−1 = 𝜙𝑙;

𝑝𝑙 (𝑥0𝐼 + 𝑥1𝜋 + ⋅ ⋅ ⋅ + 𝑥𝑛−1𝜋𝑛−1

) = 𝑥𝑙𝜋𝑙,

(11)

for 𝑙 = 0, 1, . . . , 𝑛 − 1.

Proof. Since 𝑝𝑙 = (1/𝑛)∑𝑛−1

𝑗=0𝜂−𝑗𝜔−𝑙𝑗

𝜙𝑗, we gain that

𝑝2

𝑙=

1

𝑛2[𝜙0+ 𝜂−1𝜔−𝑙𝜙 + ⋅ ⋅ ⋅ + 𝜂

−(𝑛−1)𝜔−(𝑛−1)𝑙

𝜙𝑛−1

]

2

=

1

𝑛2[𝜙0+ 𝜂−1𝜔−𝑙𝜙 + ⋅ ⋅ ⋅ + 𝜂

−(𝑛−1)𝜔−(𝑛−1)𝑙

𝜙𝑛−1

+ 𝜂−1𝜔−𝑙𝜙 + ⋅ ⋅ ⋅ + 𝜂

−𝑛𝜔−𝑛𝑙

𝜙𝑛

+ ⋅ ⋅ ⋅ + 𝜂−(𝑛−1)

𝜔−(𝑛−1)𝑙

𝜙𝑛−1

+ ⋅ ⋅ ⋅ + 𝜂−2(𝑛−1)

𝜔−2(𝑛−1)𝑙

𝜙2(𝑛−1)

]

= 𝑝𝑙.

(12)

Similarly, we can obtain 𝑝𝑙𝑝𝑗 = 0 for 𝑙 = 𝑗.And, by calculation, we have

𝑝0 + 𝑝1 + ⋅ ⋅ ⋅ + 𝑝𝑛−1

=

1

𝑛

[

[

𝑛−1

∑

𝑗=0

𝜂−𝑗𝜙𝑗+ ⋅ ⋅ ⋅ +

𝑛−1

∑

𝑗=0

𝜔−(𝑛−1)𝑗

𝜂−𝑗𝜙𝑗]

]

=

1

𝑛

[

[

𝑛−1

∑

𝑗=0

𝜂−𝑗𝜙𝑗(𝜔0+ 𝜔−𝑗

+ ⋅ ⋅ ⋅ + 𝜔−(𝑛−1)𝑗

)]

]

=

1

𝑛

[

[

𝑛−1

∑

𝑗=0

𝜂−𝑗𝜙𝑗(

1 − 𝜔−𝑛𝑗

1 − 𝜔−𝑗

)]

]

= 𝜙0,

𝜂𝑙𝑝0 + 𝜂

𝑙𝜔𝑙𝑝1 + 𝜂

𝑙𝜔2𝑙𝑝2 + ⋅ ⋅ ⋅ + 𝜂

𝑙𝜔(𝑛−1)𝑙

𝑝𝑛−1

=

1

𝑛

[

[

𝑛−1

∑

𝑗=0

𝜂𝑙𝜙𝑗+

𝑛−1

∑

𝑗=0

𝜂𝑙𝜙𝑗+ ⋅ ⋅ ⋅ +

𝑛−1

∑

𝑗=0

𝜂𝑙𝜙𝑗]

]

= 𝜙𝑙.

(13)

Finally, we gain

𝑝𝑙 (𝑥0𝐼 + 𝑥1𝜋 + 𝑥2𝜋2+ ⋅ ⋅ ⋅ + 𝑥𝑛−1𝜋

𝑛−1)

=

1

𝑛

𝑛−1

∑

𝑗=0


𝜙𝑗(𝑆)


=

1

𝑛

[𝑥0𝐼 + 𝑥1𝜋 + 𝑥2𝜋2+ ⋅ ⋅ ⋅ + 𝑥𝑛−1𝜋

𝑛−1

+ 𝜂−1𝜔−𝑙(𝜂𝑥0𝐼 + 𝜂𝜔𝑥1𝜋 + ⋅ ⋅ ⋅ + 𝜂𝜔

𝑛−1𝑥𝑛−1𝜋

𝑛−1)

+⋅ ⋅ ⋅+ 𝜂−(𝑛−1)

𝜔−(𝑛−1)𝑙

(𝜂𝑛−1

𝑥0𝐼 + 𝜂𝑛−1

𝜔𝑛−1

𝑥1𝜋

+⋅ ⋅ ⋅+ 𝜂𝑛−1

𝜔(𝑛−1)

2


)]

= 𝑥𝑙𝜋𝑙.

(14)

Synthesizing Propositions 1, 2, and 3, we get the followingtheorem.

Theorem 4. The algebras generated by 𝐼, 𝜋, 𝜋2, . . . , 𝜋

𝑛−1 and𝜙0, 𝜙1, 𝜙2, . . . , 𝜙

𝑛−1 over C are isomorphic and can be calledskew-circulant algebras.

3. Functional Equations forthe Skew-Circulant Algebra

Proposition 5. If 𝑓 is a linear entire function, 𝑓 : C → C,then

𝑓 (𝑐0𝐼 + 𝑐1𝜋 + ⋅ ⋅ ⋅ + 𝑐𝑛−1𝜋𝑛−1

)

=

𝑛−1

∑

𝑙=0

[

[

1

𝑛

𝑛−1

∑

𝑗=0


𝑓(

𝑛−1

∑

𝑘=0

𝜂𝑘𝜔𝑗𝑘𝑐𝑘)

]

]

𝜋𝑙.

(15)

Proof. According to [3], we know that

𝜆𝑙 =

𝑛−1

∑

𝑗=0

𝜂𝑗𝜔𝑙𝑗𝑐𝑗,

𝑐𝑙 =1

𝑛

𝑛−1

∑

𝑗=0


𝜆𝑗,

(16)

for 𝑙 = 0, 1, . . . , 𝑛 − 1.We thus have

𝑛−1

∑

𝑙=0

𝑓 (𝑐𝑙) 𝜋𝑙=

𝑛−1

∑

𝑙=0

𝑓(

1

𝑛

𝑛−1

∑

𝑗=0


𝜆𝑗)𝜋𝑙

=

𝑛−1

∑

𝑙=0

1

𝑛

𝑛−1

∑

𝑗=0


𝑓 (𝜆𝑗) 𝜋𝑙

=

𝑛−1

∑

𝑙=0

[

[

1

𝑛

𝑛−1

∑

𝑗=0


𝑓(

𝑛−1

∑

𝑘=0

𝜂𝑘𝜔𝑗𝑘𝑐𝑘)

]

]

𝜋𝑙.

(17)

Then, we complete the proof of this conclusion.

For any linear entire function 𝑓 : C → C, (15) can bewritten as follows:

𝑓 (𝑐0𝐼 + 𝑐1𝜋 + ⋅ ⋅ ⋅ + 𝑐𝑛−1𝜋𝑛−1

)

=

𝑛−1

∑

𝑙=0

𝐹𝑙 (𝑐0, 𝑐1, . . . , 𝑐𝑛−1) 𝜋𝑙,

(18)

where

𝐹𝑙 (𝑐0, 𝑐1, . . . , 𝑐𝑛−1)

=

1

𝑛

𝑛−1

∑

𝑘=0

𝜂−𝑘𝜔−𝑙𝑘

𝑓(

𝑛−1

∑

𝑗=0

𝜂𝑗𝜔𝑗𝑘𝑐𝑗) ,

(19)

𝑙 = 0, 1, 2, . . . , 𝑛 − 1.For each 𝑙, 𝐹0, 𝐹1, . . . , 𝐹𝑛−1 satisfy the functional equation

𝐹𝑙 (𝜂𝑐0, 𝜂𝜔𝑐1, 𝜂𝜔2𝑐2, . . . , 𝜂𝜔

𝑛−1𝑐𝑛−1)

= 𝜂𝜔𝑙𝐹𝑙 (𝑐0, 𝑐1, . . . , 𝑐𝑛−1) ,

(20)

for 𝜂 = 𝑒𝜋𝑖/𝑛, 𝜔 = 𝑒

2𝜋𝑖/𝑛, all 𝑙 = 0, 1, 2, . . . , 𝑛 − 1, and all𝑐0, 𝑐1, . . . , 𝑐𝑛−1 ∈ C.

Equation (20) is related to the skew-circulant algebra alsoin another way.

If the function 𝐹 is defined by

𝐹 (𝑐0, 𝑐1, . . . , 𝑐𝑛−1) =1

𝑛

𝑛−1

∑

𝑘=0


𝑓(

𝑛−1

∑

𝑗=0

𝜂𝑗𝜔𝑗𝑘𝑐𝑗) , (21)

let𝑈 = {𝐹 | 𝐹 : C𝑛 → C} and let𝑂 be the operator𝑂 : 𝑈 →

𝑈, linear in 𝐹, defined by

𝑂 (𝐹) (𝑐0, 𝑐1, . . . , 𝑐𝑛−1)

= 𝐹 (𝜂𝑐0, 𝜂𝜔𝑐1, 𝜂𝜔2𝑐2, . . . , 𝜂𝜔

𝑛−1𝑐𝑛−1) ;

(22)

that is,

𝑂 (𝐹) = 𝜂𝜔𝑙𝐹. (23)

If we denote𝑂𝑘 the operation𝑂 composed with itself 𝑘 times,then

𝑂𝑘(𝐹) (𝑐0, 𝑐1, 𝑐2, . . . , 𝑐𝑛−1)

= 𝐹 (𝜂𝑘𝑐0, 𝜂𝑘𝜔𝑘𝑐1, . . . , 𝜂

𝑘𝜔(𝑛−1)𝑘

𝑐𝑛−1) ,

(24)

where 𝑂𝑗= −𝑂

0 if and only if 𝑛 divides 𝑗 and 𝑗/𝑛 is an oddnumber; 𝑂𝑗 = 𝑂

0 if and only if 𝑛 divides 𝑗 and 𝑗/𝑛 is an evennumber.


All these equations (15)–(24) lead up to the followingtheorem.

Theorem 6. Linear combinations of the operators𝑂0, 𝑂1, . . . , 𝑂

𝑛−1 over C form a skew-circulant algebra.

Proposition 7. If the operators 𝑅0, 𝑅1, . . . , 𝑅𝑛−1 : 𝑈 → 𝑈 aredefined by

𝑅𝑙 =1

𝑛

(𝑂0+ 𝜂−1𝜔−𝑙𝑂1+ 𝜂−2𝜔−2𝑙

𝑂2

+ ⋅ ⋅ ⋅ + 𝜂−(𝑛−1)

𝜔−(𝑛−1)𝑙

𝑂𝑛−1

) ,

(25)

for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1, then these operators 𝑅𝑙 (𝑙 =

0, 1, 2, . . . , 𝑛 − 1) have the following properties:

𝑅2

𝑙= 𝑅𝑙, for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1; (26)

𝑅𝑙𝑅𝑗 = 0 𝑖𝑓 𝑙 = 𝑗; (27)

𝑅0 + 𝑅1 + ⋅ ⋅ ⋅ + 𝑅𝑛−1 = 𝑂0; (28)

𝑅0 + 𝜂𝜔𝑙𝑅1 + ⋅ ⋅ ⋅ + 𝜂

𝑛−1𝜔(𝑛−1)𝑙

𝑅𝑛−1 = 𝑂𝑙, (29)

for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1.

Through simple calculation, we can get properties (26)–(28). Properties (26)–(29) are similar to those of the functions𝐺0, 𝐺1, . . . , 𝐺𝑛−1 and 𝑝0, 𝑝1, . . . , 𝑝𝑛−1.

By all accounts, a function 𝐹 in 𝑈 satisfies (20) if andonly if 𝐹 ∈ Ran𝑅𝑙 (the range of the operator 𝑅𝑙 was denotedby Ran𝑅𝑙). Moreover, properties (26)–(28) above imply that𝑈 = Ran𝑅0 ⊕ Ran𝑅1 ⊕ ⋅ ⋅ ⋅ ⊕ Ran𝑅𝑛−1, each function 𝐹𝑙,𝑙 = 0, 1, 2, . . . , 𝑛 − 1, defined by (19) is in Ran𝑅𝑙.

In addition,

𝐹0 + 𝐹1 + ⋅ ⋅ ⋅ + 𝐹𝑛−1

= 𝑓 (𝑐0 + 𝜂𝑐1 + ⋅ ⋅ ⋅ + 𝜂𝑛−1

𝑐𝑛−1) .

(30)

Thus

𝐹𝑙 = 𝑅𝑙 (𝑓 (𝑐0 + 𝜂𝑐1 + ⋅ ⋅ ⋅ + 𝜂𝑛−1

𝑐𝑛−1)) . (31)

4. Other Skew-Circulant Algebras

By (25) and (24), if a function 𝑔maps C𝑛 into C, then

𝑅𝑙 (𝑔) (𝑐0, 𝑐1, . . . , 𝑐𝑛−1)

=

1

𝑛

𝑛−1

∑

𝑘=0


𝑔 (𝜂𝑘𝑐0 + 𝜂𝑘𝜔𝑐1 + 𝜂

𝑘𝜔2𝑘𝑐2 + ⋅ ⋅ ⋅

+ 𝜂𝑘𝜔(𝑛−1)𝑘

𝑐𝑛−1) ,

(32)

for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1.

For 𝑓 : 𝑆𝑛 → 𝑆𝑛, 𝑆𝑛 is the space of skew-circulantmatrices, there exist functions 𝑓0, 𝑓1, 𝑓2, . . . , 𝑓𝑛−1 : C𝑛 → Csuch that

𝑓(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) =

𝑛−1

∑

𝑙=0

𝑓𝑙 (𝑐0, 𝑐1, 𝑐2, . . . , 𝑐𝑛−1) 𝜋𝑙. (33)

Hence, from (28),

𝑓(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) =

𝑛−1

∑

𝑙=0

𝑛−1

∑

𝑖=0

𝑅𝑙+𝑖 (𝑓𝑙) 𝜋𝑙

=

𝑛−1

∑

𝑖=0

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑓𝑙) 𝜋𝑙.

(34)

Proposition 8. Let 𝑉 = {𝑓 | 𝑓 : 𝑆𝑛 → 𝑆𝑛, 𝑆𝑛 is the space ofskew-circulant matrices}, for 𝑓 ∈ 𝑉; let us define 𝑞𝑖(𝑓) and 𝑔𝑖

such that

𝑞𝑖 (𝑓)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) =

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑓𝑙) 𝜋𝑙, (35)

𝑔𝑖 =

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑓𝑙) , (36)

for 𝑖 = 0, 1, 2, . . . , 𝑛 − 1 and 𝑙 + 𝑖 taken modulo 𝑛. Then we canreceive the properties of the 𝑞𝑖:

𝑞2

𝑖= 𝑞𝑖 for 𝑖 = 0, 1, 2, . . . , 𝑛 − 1;

𝑞𝑖𝑞𝑗 = 0 𝑖𝑓 𝑖 = 𝑗;

(𝑞0 + 𝑞1 + 𝑞2 + ⋅ ⋅ ⋅ + 𝑞𝑛−1) (𝑓) = 𝑓, 𝑓 ∈ 𝑉.

(37)

From what has been discussed above, we gain the follow-ing theorem.

Theorem 9. The 𝑞0, 𝑞1, 𝑞2, . . . , 𝑞𝑛−1 are orthogonal projectionson 𝑉, adding to the identity function on 𝑉, and so generatingover C a skew-circulant algebra.

Proposition 10. The projections 𝑞𝑖(𝑓) have another formulaas follows:

𝑞𝑖 (𝑓)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) =

1

𝑛

𝑛−1

∑

𝑘=0

𝜔−𝑖𝑘

𝜙−𝑘𝑓𝜙𝑘(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) ,

𝑖 = 0, 1, . . . , 𝑛 − 1.

(38)


Proof. We can prove (38) in the following:

𝑞𝑖 (𝑓)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙)

=

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑓𝑙) 𝜋𝑙

=

𝑛−1

∑

𝑙=0

[

1

𝑛

𝑛−1

∑

𝑘=0

𝜂−𝑘𝜔−(𝑙+𝑖)𝑘

𝑓𝑙 (𝜂𝑘𝑐0, 𝜂𝑘𝜔𝑘𝑐1, . . . ,

𝜂𝑘𝜔(𝑛−1)𝑘

𝑐𝑛−1) ]𝜋𝑙

=

1

𝑛

𝑛−1

∑

𝑙=0

[

𝑛−1

∑

𝑘=0


𝜔−𝑖𝑘

𝑓𝑙 (𝜂𝑘𝑐0, . . . ,


𝑐𝑛−1) ]𝜋𝑙

=

1

𝑛

𝑛−1

∑

𝑘=0

𝜂−𝑘𝜔−𝑖𝑘𝑛−1

∑

𝑙=0

𝜔−𝑙𝑘

𝑓𝑙 (𝜂𝑘𝑐0, 𝜂𝑘𝜔𝑘𝑐1, . . . ,


𝑐𝑛−1) 𝜋𝑙

=

1

𝑛

𝑛−1

∑

𝑘=0

𝜔−𝑖𝑘

𝜙−𝑘𝑓(

𝑛−1

∑

𝑙=0

𝜂𝑘𝜔𝑙𝑘𝑐𝑙𝜋𝑙)

=

1

𝑛

𝑛−1

∑

𝑘=0

𝜔−𝑖𝑘

𝜙−𝑘𝑓𝜙𝑘(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) .

(39)

This result can be rewritten in the form

𝑞𝑖 (𝑓)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) =

1

𝑛

𝑛−1

∑

𝑘=0

𝜔−𝑖𝑘

𝜙−𝑘𝑓𝜙𝑘, (40)

𝑖 = 0, 1, . . . , 𝑛 − 1, for all functions 𝑓 ∈ 𝑉.Finally, it can be shown that 𝑓𝜙 = 𝜔

𝑖𝜙𝑓 if and only if 𝑓 ∈

Ran𝑝𝑖.

Theorem 11. For each 𝑓 ∈ 𝑉 and every 𝑖 = 0, 1, 2, . . . , 𝑛 −

1, there exists only one function 𝑔𝑖 such that the followingequation holds:

𝑞𝑖 (𝑓)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) =

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑔𝑖) 𝜋𝑙, (41)

for 𝑔𝑖 = ∑𝑛−1

𝑙=0𝑅𝑙+𝑖(𝑓𝑙), 𝑖 = 0, 1, 2, . . . , 𝑛 − 1 and 𝑙 + 𝑖 taken

modulo 𝑛.

Proof. By (26) and (27),

𝑅𝑙+𝑖 (𝑔𝑖) = 𝑅𝑙+𝑖 (𝑓𝑙) (42)

for 𝑙 = 0, 1, 2, . . . , 𝑛 − 1; 𝑖 = 0, 1, 2, . . . , 𝑛 − 1 and 𝑙 + 𝑖 takenmodulo 𝑛.

Suppose there exists another function 𝑔∗

𝑖: C𝑛 → C

such that 𝑞𝑖(𝑓) = ∑𝑛−1

𝑙=0𝑅𝑙+𝑖(𝑔𝑖)𝜋

𝑙= ∑𝑛−1

𝑙=0𝑅𝑙+𝑖(𝑔

∗

𝑖)𝜋𝑙. Since

𝐼, 𝜋, 𝜋2, . . . , 𝜋

𝑛−1 is a basis of 𝑆𝑛, 𝑅𝑙+𝑖(𝑔𝑖) = 𝑅𝑙+𝑖(𝑔∗

𝑖) for 𝑙 =

0, 1, . . . , 𝑛 − 1. By (28),

𝑔𝑖 =

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑔𝑖) =

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑔∗

𝑖) = 𝑔∗

𝑖. (43)

So 𝑔𝑖 is unique. Thus, there is an isomorphism betweenfunctions 𝑔𝑖 : C𝑛 → C and functions ∑

𝑛−1

𝑙=0𝑅𝑙+𝑖(𝑔𝑖)𝜋

𝑙 inRan𝑞𝑖.

Now the results of all this are as follows: let Φ = {𝑓 | 𝑓 :

C → C} with 𝑓 an entire function and 𝑈 = {𝑓 | 𝑓 : C𝑛 →

C}. Let 𝐼 be a monomorphism Φ → 𝑈𝑛 defined by 𝐼(𝑓) =

(𝑓(𝑐0+𝜂𝑐1+⋅ ⋅ ⋅+𝜂𝑛−1

𝑐𝑛−1), 0, . . . , 0). Let 𝜏 be amonomorphismΦ → 𝑉 defined by

𝜏 (𝑓)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙)

=

𝑛−1

∑

𝑙=0

𝑅𝑛 (𝑓 (𝑐0 + 𝜂𝑐1 + ⋅ ⋅ ⋅ + 𝜂𝑛−1

𝑐𝑛−1)) 𝜋𝑙

= 𝑓(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙) ,

(44)

which follows from (18), (19), and (31). Then there exists anisomorphism 𝜃 : 𝑈

𝑛→ 𝑉 defined by

𝜃 (𝑔0, 𝑔1, 𝑔2, . . . , 𝑔𝑛−1)(

𝑛−1

∑

𝑙=0

𝑐𝑙𝜋𝑙)

=

𝑛−1

∑

𝑖=0

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑔𝑖) 𝜋𝑙.

(45)

5. A Linear Involution

Suppose that 𝑔 is a function 𝑆𝑛 → 𝑆𝑛 given by

𝑔 =

𝑛−1

∑

𝑖=0

𝑔𝑖𝜋𝑖, (46)

where 𝑔𝑖 is given by (36). Written out, we have

𝑔 =

𝑛−1

∑

𝑖=0

[

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑓𝑙)] 𝜋𝑖. (47)

Let 𝑉 denote the space of functions 𝑓 : 𝑆𝑛 → 𝑆𝑛. If𝑓 is an element of 𝑉, then there exist a set of 𝑛 functions𝑓0, 𝑓1, . . . , 𝑓𝑛−1 mapping C𝑛 into C such that 𝑓 = ∑

𝑛−1

𝑙=0𝑓𝑙𝜋𝑙

(like (33)). Let us switch the 𝑙 and the 𝑖 in the right-hand sideof (47). Then let 𝜓 be the function 𝑉 → 𝑉 defined by

𝜓(

𝑛−1

∑

𝑙=0

𝑓𝑙𝜋𝑙) =

𝑛−1

∑

𝑙=0

(

𝑛−1

∑

𝑖=0

𝑅𝑙+𝑖 (𝑓𝑖))𝜋𝑙. (48)


Theorem 12. From (26), (27), and (28), we can show that 𝜓 isa linear involution on 𝑉.

Proof. From (26), (27), and (28), we get

𝜓(𝜓(

𝑛−1

∑

𝑙=0

𝑓𝑙𝜋𝑙)) =

𝑛−1

∑

𝑙=0

{

𝑛−1

∑

𝑖=0

𝑅𝑙+𝑖 [

𝑛−1

∑

𝑘=0

𝑅𝑖+𝑘 (𝑓𝑘)]}𝜋𝑙

=

𝑛−1

∑

𝑙=0

[

𝑛−1

∑

𝑖=0

𝑅𝑙+𝑖 (𝑓𝑙)] 𝜋𝑙

=

𝑛−1

∑

𝑙=0

𝑓𝑙𝜋𝑙.

(49)

That is, 𝜓2 = 𝜓0 (the identity function on 𝑉). Then, 𝜓 is a

linear involution on 𝑉.

Consider the set𝑊 = {𝛼𝜓0+𝛽𝜓 | 𝛼, 𝛽 ∈ C}, that is, linear

combinations over C of 𝜓0 and 𝜓 (since 𝜓2 = 𝜓0). Then𝑊 is

a 2×2 complex circulant algebra; (𝜓0+𝜓)/2 and (𝜓0−𝜓)/2 are

idempotent elements of𝑊; that is, they are projections on 𝑉.If 𝑓 is in 𝑉, then 𝜓(𝑓) = 𝑓 if and only if 𝑓 ∈ Ran(𝜓0 + 𝜓)/2,and 𝜓(𝑓) = −𝑓 if and only if 𝑓 ∈ Ran(𝜓0 − 𝜓)/2. Also, 𝑉 =

(Ran(𝜓0 + 𝜓)/2) ⊕ (Ran(𝜓0 − 𝜓)/2) (a direct sum).

Example 13. If 𝑛 = 2, then 𝜋2

= −𝐼. Let 𝑓 and 𝑔 be twofunctions C2 → C. Then

𝜓 (𝐼𝑓 (𝑐0, 𝑐1) + 𝜋𝑔 (𝑐0, 𝑐1))

=

𝐼

2

[𝑓 (𝑐0, 𝑐1) − 𝑖𝑓 (𝑖𝑐0, −𝑖𝑐1) + 𝑔 (𝑐0, 𝑐1) + 𝑖𝑔 (𝑖𝑐0, −𝑖𝑐1)]

+

𝜋

2

[𝑓 (𝑐0, 𝑐1) + 𝑖𝑓 (𝑖𝑐0, −𝑖𝑐1)

+ 𝑔 (𝑐0, 𝑐1) − 𝑔 (−𝑐0, −𝑐1)] .

(50)

Note that, if 𝑓𝑖 is a function C𝑛 → C for each 𝑖, we haveby (48) that

𝜓 (𝑓𝑖𝜋𝑖) =

𝑛−1

∑

𝑙=0

𝑅𝑙+𝑖 (𝑓𝑖) 𝜋𝑙. (51)

In the same way, (48) implies that

𝜓 [𝑅𝑙+𝑖 (𝑓𝑙) 𝜋𝑙] = 𝑅𝑙+𝑖 (𝑓𝑙) 𝜋

𝑖, (52)

where 𝑙 and 𝑖 vary from 0 to 𝑛 − 1 and 𝑙 + 𝑖 is taken modulo 𝑛.Since𝜓(𝑓) = 𝑓 if and only if𝑓 ∈ Ran(𝜓0+𝜓)/2, the function

(

1

2

) (𝜓0+ 𝜓) [𝑅𝑙+𝑖 (𝑓𝑙) 𝜋

𝑙]

= (

1

2

)𝑅𝑙+𝑖 (𝑓𝑙) (𝜋𝑙+ 𝜋𝑖)

(53)

is a fixed point of 𝜓.

6. Conclusion

We prove that the set of skew-circulants with complex entrieshas an idempotent basis.This paper displays algebras of oper-ators which are isomorphic to the algebra of 𝑛 × 𝑛 complexskew-circulant matrices. In [19], a new fast algorithm foroptimal design of block digital filters (BDFs) is proposedbased on skew circulant matrix. The reason why we focusour attention on skew-circulant operator is to explore theapplication of skew-circulant in the related field. On the basisof existing application situation [2–8], we will exploit solvingordinary, partial, and delay differential equations based onskew circulant operator.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The research is supported by the Development Projectof Science & Technology of Shandong Province (Grantno. 2012GGX10115) and NSFC (Grant no. 11301251) and theAMEP of Linyi University, China.

References

[1] D. Bertaccini and M. K. Ng, “Skew-circulant preconditionersfor systems of LMF-based ODE codes,” in Numerical Analysisand Its Applications, vol. 1988 of Lecture Notes in ComputerScience, pp. 93–101, Springer, Berlin, Germany, 2001.

[2] J. Delgado, N. Romero, A. Rovella, and F. Vilamajo, “Boundedsolutions of quadratic circulant difference equations,” Journal ofDifference Equations and Applications, vol. 11, no. 10, pp. 897–907, 2005.

[3] A. C. Wilde, “Differential equations involving circulant matri-ces,” Rocky Mountain Journal of Mathematics, vol. 13, no. 1, pp.1–13, 1983.

[4] B. Karasozen and G. Simsek, “Energy preserving integration ofbi-Hamiltonian partial differential equations,” Applied Mathe-matics Letters, vol. 26, no. 12, pp. 1125–1133, 2013.

[5] E. W. Sachs and A. K. Strauss, “Efficient solution of a partialintegro-differential equation in finance,” Applied NumericalMathematics, vol. 58, no. 11, pp. 1687–1703, 2008.

[6] R.W. Brockett and J. L.Willems, “Discretized partial differentialequations: examples of control systems defined on modules,”Automatica, vol. 10, no. 5, pp. 507–515, 1974.

[7] R. D. Nussbaum, “Circulant matrices and differential-delayequations,” Journal of Differential Equations, vol. 60, no. 2, pp.201–217, 1985.

[8] X.-Q. Jin, S.-L. Lei, and Y.-M. Wei, “Circulant preconditionersfor solving differential equations with multidelays,” Computers&Mathematics with Applications, vol. 47, no. 8-9, pp. 1429–1436,2004.

[9] P. J. Davis, Circulant Matrices, John Wiley & Sons, New York,NY, USA, 1979.

[10] Z. L. Jiang and Z. X. Zhou, Circulant Matrices, Chengdu Tech-nology University, Chengdu, China, 1999.


[11] R. H. Chan and X.-Q. Jin, “Circulant and skew-circulantpreconditioners for skew-Hermitian type Toeplitz systems,”BITNumerical Mathematics, vol. 31, no. 4, pp. 632–646, 1991.

[12] R. H. Chan and K.-P. Ng, “Toeplitz preconditioners for Hermi-tian Toeplitz systems,” Linear Algebra and Its Applications, vol.190, pp. 181–208, 1993.

[13] T. Huckle, “Circulant and skewcirculant matrices for solvingToeplitz matrix problems,” SIAM Journal on Matrix Analysisand Applications, vol. 13, no. 3, pp. 767–777, 1992.

[14] J. N. Lyness and T. Sørevik, “Four-dimensional lattice rulesgenerated by skew-circulant matrices,”Mathematics of Compu-tation, vol. 73, no. 245, pp. 279–295, 2004.

[15] H. Karner, J. Schneid, and C. W. Ueberhuber, “Spectral decom-position of real circulant matrices,” Linear Algebra and ItsApplications, vol. 367, pp. 301–311, 2003.

[16] M. J. Narasimha, “Linear convolution using skew-cyclic convo-lutions,” IEEE Signal Processing Letters, vol. 14, no. 3, pp. 173–176,2007.

[17] V. C. Liu and P. P. Vaidyanathan, “Circulant and skew-circulantmatrices as new normal-form realization of IIR digital filters,”IEEE Transactions on Circuits and Systems, vol. 35, no. 6, pp.625–635, 1988.

[18] J. Li, Z. L. Jiang, N. Shen, and J.W. Zhou, “On optimal backwardperturbation analysis for the linear system with skew circulantcoefficient matrix,” Computational and Mathematical Methodsin Medicine, vol. 2013, Article ID 707381, 7 pages, 2013.

[19] D. Q. Fu, Z. L. Jiang, Y. F. Cui, and S. T. Jhang, “A new fastalgorithm for optimal design of block digital filters by skew-cyclicconvolution,” IET Signal Processing.

[20] Y. Gao, Z. L. Jiang, and Y. P. Gong, “On the determinantsand inverses of skew circulant and skew left circulant matriceswith Fibonacci and Lucas Numbers,” WSEAS Transactions onMathematics, vol. 12, no. 4, pp. 472–481, 2013.

[21] A. C. Wilde, “Solutions of equations containing primitive rootsof unity,” Journal of Undergraduate Mathematics, vol. 3, pp. 25–28, 1971.

[22] A. C.Wilde, “Cauchy-Riemann conditions for algebras isomor-phic to the circulant algebra,” The Journal of the University ofKuwait (Science), vol. 14, no. 2, pp. 189–204, 1987.

[23] A. C. Wilde, “Commutative projection operators,” Atti delSeminarioMatematico e Fisico dell’Universita diModena, vol. 35,no. 1, pp. 167–172, 1987.

[24] A. C. Wilde, “Algebras of operators isomorphic to the circulantalgebra,” Proceedings of the American Mathematical Society, vol.105, no. 4, pp. 808–816, 1989.

[25] R. H. Chan, X.-Q. Jin, andM.-C. Yeung, “The circulant operatorin the banach algebra of matrices,” Linear Algebra and ItsApplications, vol. 149, pp. 41–53, 1991.

[26] D. Chillag, “Regular representations of semisimple algebras,separable field extensions, group characters, generalized cir-culants, and generalized cyclic codes,” Linear Algebra and ItsApplications, vol. 218, pp. 147–183, 1995.

[27] C. Brink and J. Pretorius, “Boolean circulants, groups, andrelation algebras,”TheAmericanMathematical Monthly, vol. 99,no. 2, pp. 146–152, 1992.

[28] R. H. Chan, X.-Q. Jin, and M. K. Ng, “Circulant integral oper-ators as preconditioners for Wiener-Hopf equations,” IntegralEquations and Operator Theory, vol. 21, no. 1, pp. 12–23, 1995.

[29] F. di Benedetto and S. serra Capizzano, “Optimal multilevelmatrix algebra operators,” Linear and Multilinear Algebra, vol.48, no. 1, pp. 35–66, 2000.

[30] F. di Benedetto and S. serra Capizzano, “A note on the superop-timal matrix algebra operators,” Linear andMultilinear Algebra,vol. 50, no. 4, pp. 343–372, 2002.

[31] I. S. Hwang, D.-O. Kang, and W. Y. Lee, “Hyponormal Toeplitzoperators with matrix-valued circulant symbols,” ComplexAnalysis and Operator Theory, vol. 7, no. 4, pp. 843–861, 2013.

[32] W. Bani-Domi and F. Kittaneh, “Norm equalities and inequali-ties for operator matrices,” Linear Algebra and Its Applications,vol. 429, no. 1, pp. 57–67, 2008.

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