eigensolution of specially structured matrices with hyper-symmetry

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2006; 67:1012–1043Published online 9 February 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1660

Eigensolution of specially structured matrices withhyper-symmetry

A. Kaveh∗,† and M. A. Sayarinejad

Iran University of Science and Technology, Narmak, Tehran-16, Iran

SUMMARY

Recently, four canonical forms have been developed and applied to the dynamics and stability analysisof symmetric frames. In this paper, hyper-symmetric matrices and specially structured matrices aredefined and efficient methods are proposed for the eigensolution of such matrices. Applications areextended to hyper-graphs and specially structured graphs. Simple methods are developed for calculatingthe eigenvalues of the Laplacian matrices of such graphs. The developments presented in this papercan also be considered as generalization of Form II and Form III symmetry, previously defined bythe authors. Copyright � 2006 John Wiley & Sons, Ltd.

KEY WORDS: hyper-symmetric matrices; specially structured matrices; hyper-symmetric graphs;specially structured graphs; factorization tree; Laplacian; eigenvalues

1. INTRODUCTION

The evaluation of the characteristic roots of a given n-square matrix requires computation of annth-order determinant and the roots of an equation of nth degree-formidable, often impossible,tasks. There are, however, classes of matrices whose characteristic roots can be calculatedmore easily by factorization techniques. Some of these matrices are previously studied [1–4],and here some other compound matrices with canonical forms are introduced for which theeigensolutions can be obtained much simpler compared to the classic approaches.

Symmetry has been widely studied in science and engineering [5–9]. Eigenvalue problemsarise in many scientific and engineering problems [10–13]. While the basic mathematicalideas are independent of the size of matrices, the numerical determination of eigenvalues andeigenvectors requires additional consideration as the dimensions and the sparsity of matricesincrease. Special methods are needed for efficient solution of such problems.

∗Correspondence to: A. Kaveh, Iran University of Science and Technology, Narmak, Tehran-16, Iran.†E-mail: [email protected]

Received 29 June 2005Revised 24 November 2005

Copyright � 2006 John Wiley & Sons, Ltd. Accepted 6 January 2006

EIGENSOLUTION OF SPECIALLY STRUCTURED MATRICES 1013

Methods are developed for decomposing the graph models of structures in order to calculatethe eigenvalues of matrices with special patterns, References [1–4]. The eigenvectors corre-sponding to such patterns are studied in Reference [2]. The application of these methods isextended to the vibration of mass–spring systems [14], and free vibration of frames [15].

In this paper, hyper-symmetric matrices and specially structured matrices are defined andefficient methods are proposed for eigensolution of such matrices. Applications are extendedto hyper-graphs and specially structured graphs. Simple methods are developed for calculatingthe eigenvalues of the Laplacian matrices of such graphs. The content of this paper can alsobe considered as generalization of Form II and Form III symmetry, previously defined by theauthors [1–3].

2. MATRICES WITH CANONICAL FORMS

Here, an N × N symmetric matrix [M] is considered with all entries being real. For three specialcanonical forms, the eigenvalues of [M] are obtained using the properties of its submatrices.

Canonical Form I: In this case [M] has the following pattern:

[M] =⎡⎣[A]n×n [0]n×n

[0]n×n [A]n×n

⎤⎦

N×N

(1)

with N = 2n.Considering the set of eigenvalues of the submatrix [A] as {�A}, the set of eigenvalues of

[M] can be obtained as

{�M} = {�A}∪{�A} (2)

Since det M = det A × det A, the above relation becomes obvious. The sign ∪ simply indicatesthe collection of the eigenvalues of the submatrices.Canonical Form II: For this case, matrix [M] can be decomposed into the following form:

[M] =⎡⎣[A]n×n [B]n×n

[B]n×n [A]n×n

⎤⎦

N×N

(3)

The eigenvalues of [M] can be calculated as

{�M} = {�C}∪̄{�D} (4)

where

[C] = [A] + [B] and [D] = [A] − [B] (5)

[C] and [D] are called condensed submatrices of [M]. The proof can be found inReferences [2, 14].

Copyright � 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 67:1012–1043

1014 A. KAVEH AND M. A. SAYARINEJAD

Canonical Form III: This form has a Form II submatrix augmented by some rows and columns,shown as follows:

[M] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[A] [B] L11

L21

· · ·· · ·

L1k

L2k

[B] [A]

Ln1

L11

L21

Ln1

· · ·· · ·· · ·· · ·

Lnk

L1k

L2k

Lnk

C(2n + 1, 1) · · · C(2n + 1, 2n) C(2n + 1, 2n + 1) · · · C(2n + 1, 2n + k)...

......

......

...

Z(2n + k, 1) · · · Z(2n + k, 2n) Z(2n + k, 2n + 1) · · · Z(2n + k, 2n + k)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(6)

where [M] is a (2n + k) × (2n + k) matrix, with a 2n × 2n submatrix with the pattern ofForm II, and k augmented columns and rows. The entries of the augmented columns are thesame in each column, and all the entries of [M] are real numbers.

The set of eigenvalues for [M] is obtained as

{�M} = {�D}∪̄{�E} (7)

where D and E are constructed as follows:

D=[A] − [B] (8a)

[E]=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

L11

L21

· · ·· · ·

L1k

L2k

[A + B]...

Ln1

· · ·· · ·

...

Lnk

C(2n + 1, 1) + C(2n + 1, n + 1) · · · C(2n + 1, 2n + 1) · · · C(2n + 1, 2n + k)

......

......

...

Z(2n + k, 1) + Z(2n + k, n + 1) · · · Z(2n + k, 2n + 1) · · · Z(2n + k, 2n + k)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8b)

For the proof the reader may refer to References [2, 14].



3. COMMUTATIVITY OF MATRICES

3.1. Commutativity property of matrices

In general, the product of two matrices A and B is not commutative, i.e. AB �= BA. However,this property holds for some case as follows:

(a) For a 2 × 2 matrices as

A =⎡⎣a b

b a

⎤⎦ and B =

⎡⎣a′ b′

b′ a′

⎤⎦ ⇒ A · B = B · A (9)

where A and B have Form II symmetry.(b) For a 2 × 2 matrices of the following form:

A =⎡⎣a −b

b a

⎤⎦ and B =

⎡⎣a′ −b′

b′ a′

⎤⎦ ⇒ A · B = B · A (10)

(c) If one of the matrices is scalar,

S =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

K 0 · · 0

· K

· ·· ·0 K

⎤⎥⎥⎥⎥⎥⎥⎥⎦

= KI (11)

then for an arbitrary matrix AN×N the following holds:

A × S = S × A = KA (12)

(d) Both matrices are diagonal.For the matrices which commute, the determinant, eigenvalues and eigenvectors can

be determined, since the characteristic equation can easily be factorized.As an example, the following matrix is considered and its determinant and eigenvalues

are calculated. Now if two blocks A and C are commutative, then the relationships fordetermining the determinant and eigenvalues are obtained.

M =⎡⎣A B

C D

⎤⎦

2n×2n

⇒

⎡⎢⎢⎢⎢⎢⎣

det[M] = AD − BC

eig[M] =

⎡⎢⎢⎣

�1 = A + D2

+ 1

2

√(A − D)2 + 4BC

�2 = A + D2

− 1

2

√(A − D)2 + 4BC

(13)

For any other M which can be decomposed into n × n blocks having the commutativityproperty, the determinant and eigenvalues can easily be obtained.



3.2. Commutativity of Form II matrices with higher dimensions

Suppose two matrices with Form II symmetry have blocks which have also Form II symmetry.Then these two matrices will also be commutative. Such matrices are called specially structuredmatrices.

4. HYPER-SYMMETRIC MATRICES

Square matrices Mm×m together with two matrix operations + and × form a ring(Mm×m, +, × ), Reference [16]. Consider M as the set of matrices with Form II symmetry,i.e.

∀m2n×2n ∈ M ⇒ m =[

A B

B A

](14)

then the following properties hold:

1. The set M is closed with respect to addition

∀m, m′ ∈ M ⇒ m =[

A B

B A

]and m′ =

[A′ B′

B′ A′

](15)

m + m′ =⎡⎣A + A′ B + B′

B + B′ A + A′

⎤⎦ (16)

If

[A + A′ = A′′

B + B′ = B′′ then m + m′ = M′′ =[

A′′ B′′

B′′ A′′

]∈ M (17)

2. Since (Mm×m, +, × ) is a group, therefore the square matrices with respect to additionis associative and thus the associativity property holds for all the elements of the set M.

3. The neutral element with respect to addition is a null matrix, 0, which itself has Form IIsymmetry.

4. The inverse of each element of M, with respect to addition, is the same element withreverse sign, which is also in the set.

∀m2n×2n ∈ M ∃−m2n×2n : (m2n×2n)+(−m2n×2n)=(−m2n×2n) + (m2n×2n) = 02n×2n (18)

m =[

A B

B A

]and − m =

[−A −B

−B −A

]∈ M (19)



Thus (M, +) is a group. Since

∀m2n×2n and m′2n×2n ∈ M ⇒ m + m′ = m′ + m (20)

Therefore, (M, +) is an Abelian group.

m =[

A B

B A

]and m′ =

[A′ B′

B′ A′

](21)

m × m′ =[

A B

B A

]×[

A′ B′

B′ A′

]=⎡⎣AA′ + BB′ AB′ + BA′

BA′ + AB′ BB′ + AA′

⎤⎦ (22)

If

[AA′ + BB′ = A′′

AB′ + BA′ = B′′ ⇒ m × m′ = m′′ =[

A B

B A

]×[

A′′ B′′

B′′ A′′

]∈ M (23)

Thus the set M is closed with respect to multiplication, and since (Mm×m, +, × ) is a ring,therefore the commutativity and associativity properties hold with respect to multiplicationand addition, respectively.

RemarkIt can be concluded that the set of matrices of Form II symmetry with respect to addition andmultiplication of matrices, form a ring.

Among the elements of the set M, those with n = 1 are as follows:

m2n×2n ∈ M; if n = 1 ⇒ m2×2 =[a b

b a

]2×2

(24)

It can be shown that the set of the above matrices (M2×2, +, × ) form a commutative ring.

∀m2×2 and m′2×2 ⇒ m × m′ = m′ × m (25)

i.e. for

m =[

a b

b a

]and m′ =

[a′ b′

b′ a′

](26)

we have

m × m′ =[a b

b a

]×[a′ b′b′ a′

]=⎡⎣aa′ + bb′ ab′ + ba′

ba′ + ab′ bb′ + aa′

⎤⎦ (27)

m′ × m =[a′ b′b′ a′

]×[a b

b a

]=⎡⎣a′a + bb′ a′b + b′a

b′a + a′b b′b + a′a

⎤⎦ (28)



Therefore

m × m′ = m′ × m (29)

It can be proved that if (R, ∗, 0) is a commutative ring, then all the identities and Newtonbinomial expansion hold for its elements. Therefore,

R1 =⎧⎨⎩⎡⎣a b

b a

⎤⎦

2×2

, +, ×⎫⎬⎭

forms a ring with the above properties.It is easy to show that R2 is also closed with respect to the matrix addition and multiplication

R2 =⎧⎨⎩m4×4 =

⎡⎣a b

b a

⎤⎦ : a, b ∈ R1

⎫⎬⎭ (30)

and therefore (R2, +, × ) is a commutative ring.(R3, +, × ) is a commutative ring.

R3 =⎧⎨⎩m8×8 =

⎡⎣a b

b a

⎤⎦ : a, b ∈ R2

⎫⎬⎭ (31)

∀A, B ∈ R3 ⇒ A × B = B × A (32)

As an example, the following matrix m, containing A and B as its blocks, belong to R3.

m8×8 =⎡⎣A B

B A

⎤⎦

8×8

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 5 3 7 1 2 3 45 2 7 3 2 1 4 3

3 7 2 5 3 4 1 27 3 5 2 4 3 2 1

1 2 3 4 2 5 3 72 1 4 3 5 2 7 3

3 4 4 2 3 7 2 54 3 2 1 7 3 5 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Similarly Rn with respect to matrix addition and multiplication, forms a commutative ring.

Rn =⎧⎨⎩⎡⎣a b

b a

⎤⎦

2n×2n

|a, b ∈ Rn−1

⎫⎬⎭ (33)

The matrices of Ri sets, are called hyper-symmetric matrices. The dimension of the matricesbelonging to Ri is 2i × 2i . As an example,

R4 =⎧⎨⎩⎡⎣a b

b a

⎤⎦

24×24

|a, b ∈ R3

⎫⎬⎭ (34)



and in general we have

∀m1, m2 ∈ Ri : m1 × m2 = m2 × m1 (35)

Thus hyper-symmetric matrices are commutative with respect to matrix multiplication. All hyper-symmetric matrices for commutative rings with respect to matrix addition and multiplication.

5. SPECIALLY STRUCTURED MATRICES

Consider MK×K with the following property:

MK×K = [mij ] : mij ∈ Rn (36)

This matrix is called a specially structured matrix.For this type of matrices, determinants and eigenvalues can be obtained by substituting block

matrices in place of algebraic entries in the corresponding formulae. In this way, eigendeter-minant and eigenmatrices are constructed.

As an example, the following two forms are considered:

(a) Consider matrices of the form

MK×K =[

A B

B C

]: A, B, C ∈ Rn and (K = 2i+1) (37)

Determinant of M can be calculated as

det[M] = [AC − B2] (38)

and the eigenvalues are

eig[M] =

⎡⎢⎢⎢⎣

�1 =[

A + C2

+ 1

2

√(A − C)2 + 4B2

]2i×2i

�2 =[

A + C2

− 1

2

√(A − C)2 + 4B2

]2i×2i

(39)

(b) Consider matrices of the following form:

MK×K =[

A B

C D

]: A, B, C, D ∈ Rn and (K = 2i+1) (40)

Determinant of M can be calculated as

det[M] = [AD − BC] (41)

and the eigenvalues are

eig[M] =

⎡⎢⎢⎢⎣

�1 =[

A + D2

+ 1

2

√(A − D)2 + 4BC

]2i×2i

�2 =[

A + D2

− 1

2

√(A − D)2 + 4BC

]2i×2i

(42)



M2i×2i

C D

CC CD DC DD

CCC

CCD

CDC

CDD DDC

DDDDCD

DCC

R1

R2

R3

Ri

Figure 1. The factorization tree of M2i×2i .

RemarkIf S is a set of hyper-symmetric matrices, it can be shown that this set form a commutativering with respect to matrix addition and multiplication. Therefore, the elements of this setare also closed with respect to addition and multiplication. Thus the determinant matrix andeigenmatrices are themselves hyper-symmetric.

If the matrix M2i×2i is an element of the ring Ri , the tree shown in Figure 1 can leadto the formation of 1 × 1 matrices. This tree is called a factorization tree consisting of icontours and in the ith contour the constructed matrices are 1 × 1 matrices. As an example,a M8×8 hyper-symmetric matrix, in i = 3 contour the matrices are 1 × 1 and the roots of thecharacteristic equation are easily obtainable. In the third contour of this tree, the condensedcores are the eigenvalues of the matrix M8×8.|

6. EIGENVALUES OF HYPER-SYMMETRIC MATRICES

Since hyper-symmetric matrices have Form II symmetry, therefore their condensed cores havealso Form II symmetry. A sequential decomposition of such a matrix leads to the formationof 1 × 1 matrices which are the same as the roots of the characteristic equation of the matrix.

The constituting submatrices of a hyper-symmetric matrix are 2 × 2 matrices of Form II,and their first two rows are decomposable.

Sequential condensations of hyper-symmetric matrices are in fact operations on these cores.Considering the superposition property studied in Reference [17], the following relationships

hold:

m1 =[

a0 b0

b0 a0

]and m2 =

[a1 b1

b1 a1

]and m1 + m2 = m′ (43)

eig(m1) =[

a0 + b0 = �C(m1)

a0 − b0 = �D(m1)

and eig(m2) =[

a1 + b1 = �C(m2)

a1 − b1 = �D(m2)

(44)



DDDCCD

DC

CC

M4×4

R2

R1

Figure 2. Factorization tree of M4×4.

resulting in

eig(m′) =[

�C(m1) + �C(m2) = �C(m′)

�D(m1) + �D(m2) = �D(m′)(45)

Considering the above facts, the eigenvalues of a hyper-symmetric matrix corresponds to the2 × 2 matrices of the first two rows of the matrix are obtained after appropriate partitioning.

As an example

m2×2 ∈ R1 → m2×2 =[

a0 b0

b0 a0

]⇒

[�C0 = a0 + b0

�D0 = a0 − b0(46)

m4×4 ∈ R2 → m4×4 =

⎡⎢⎢⎢⎢⎢⎣

a0 b0 a1 b1

b0 a0 b1 a1

a1 b1 a0 b0

b1 a1 b0 a0

⎤⎥⎥⎥⎥⎥⎦

4×4

=⎡⎣A0 A1

A1 A0

⎤⎦ (47)

It can be observed that the 2 × 2 blocks of the matrix m in the first rows can be identified asA0 and A1. Further decomposition leads to the eigenvalues of the matrix m.

C = A0 + A1 =⎡⎣a0 + a1 b0 + b1

b0 + b1 a0 + a1

⎤⎦

2×2

, D = A0 − A1 =⎡⎣a0 − a1 b0 − b1

b0 − b1 a0 − a1

⎤⎦

2×2

(48)

The condensed matrices C and D have Form II symmetry as

CC = [a0 + a1 + b0 + b1] = [a0 + b0] + [a1 + b1] = �C0 + �C1

CD = [a0 + a1 − b0 − b1] = [a0 − b0] + [a1 − b1] = �D0 + �D1

DC = [a0 − a1 + b0 − b1] = [a0 + b0] − [a1 + b1] = �C0 − �C1

DD = [a0 − a1 − b0 + b1] = [a0 − b0] − [a1 − b1] = �D0 − �D1

(49)

It can be observed that the eigenvalues of m are obtainable from combination of its constituting2 × 2 submatrices. The factorization tree is illustrated in Figure 2.

As it was mentioned before, for matrices belonging to R2, 1 × 1 matrices are obtained inthe second contour which are the roots of the characteristic equation.



DDDDCDDCC

R1

R2

R3

M8×8

(C1)

(C2)

(C3)

C D

CC CD DC DD

CCC CCD CDC CDD DDC

Figure 3. Factorization tree M8×8.

Consider M and construct its condensed cores:

M =[

A0 A1A1 A0

]4×4

(50)

CC = �C0 + �C1, DC = �C0 − �C1

CD = �D0 + �D1, DD = �D0 − �D1 (51)

(a) The last contour in the matrix cores constitutes the first index of �. As an example,DC indicates that the combination of �c’s have been the main purpose. The remainingcharacters from left to right indicate the notation of � in the combination. As an ex-ample, consider CC. The last character indicates that the combination of �c’s have beenconsidered.

(b) The character after C shows that the entries of the 2 × 2 matrices corresponding to theaxis of symmetry are summed in the first two rows, i.e. A0 + A1. Therefore �c’s arecombined as follows:

CC = �C0 + �C1 (52)

As an example, consider a hyper-symmetric matrix M8×8 ∈ R3. As mentioned before, thethird contour of the factorization tree corresponding to hyper-symmetric matrices containsthe eigenvalues of the matrix. Now the cores of the third contour are calculated (Figure 3).

The first two rows of a hyper-symmetric matrix of dimension 8 × 8, after being parti-tioned into 2 × 2 blocks are as follows:[

a0 b0 a1 b1 a2 b2 a3 b3

b0 a0 b1 a1 b2 a2 b3 a3

]2×8

= [A0, A1|A2, A3]2×8 (53)

As an example, CCC is calculated. The last character shows that a combination of�c is being considered. The first and second characters represent the following



operations:

First C Second C

↓ ↓A0 + A2, A1 + A3 → A0 + A2 + A1 + A3

(54)

i.e. combination of all the parts.

CCC = �C0 + �C1 + �C2 + �C3 (55)

Similarly, for the other cores we have:

CCC = �C0 + �C1 + �C2 + �C3, DCC = �C0 + �C1 − �C2 − �C3

CCD = �D0 + �D1 + �D2 + �D3, DCD = �D0 + �D1 − �D2 − �D3

CDC = �C0 − �C1 + �C2 − �C3, DDC = �C0 − �C1 − �C2 + �C3

CDD = �D0 − �D1 + �D2 − �D3, DDD = �D0 − �D1 − �D2 + �D3

(56)

7. APPLICATIONS IN THE THEORY OF GRAPHS

DefinitionA graph Gn with n nodes is called a hyper-graph if the corresponding Laplacian matrix Lbelongs to the ring Ri , i.e.

Ln×n ∈ Ri ⇒ n = 2i (57)

As an example, for R1 containing 2 × 2 Form II symmetric matrices, the following graphs aredefined (Figure 4):

L(G1) =⎡⎣ 1 −1

−1 1

⎤⎦

2×2

, L(G2) =⎡⎣ 3 −2

−2 3

⎤⎦

2×2

, L(G3) =⎡⎣4 0

0 4

⎤⎦

2×2

It can be observed that all the above graphs are symmetric.Now, graph G1 is considered as the constituting part of the hyper-graphs of higher dimen-

sions. The following graphs are provided for R2, R3 and R4 (Figure 5):Consider a Laplacian matrix of the following form:

L(G) =

⎡⎢⎢⎢⎢⎣

2 −1 −1 0

−1 2 0 −1

−1 0 2 −1

0 −1 −1 2

⎤⎥⎥⎥⎥⎦

4×4

⇒ L(G) ∈ R2



1

2

1

2

1

2

G1 G2 G3

Figure 4. Three symmetric graphs.

1

2

3

4

Figure 5. A hyper-graph G.

All the above graphs are hyper-symmetric (Figure 6). It can be seen that the graph G2 is sym-metric combination of G1 with itself. Similarly, G3 is obtained by the symmetric combinationof G2, and G4 is formed by symmetric combination of G3, etc.

The Laplacian matrices of the graphs G1–G3 are provided in the following:

L(G1) =⎡⎣ 1 −1

−1 1

⎤⎦

2×2

⇒ L(G1) ∈ R1

L(G2) =

⎡⎢⎢⎢⎢⎢⎣

2 −1 −1 0

−1 2 0 −1

−1 0 2 −1

0 −1 −1 2

⎤⎥⎥⎥⎥⎥⎦

4×4

⇒ L(G2) ∈ R2



1

22

1

3

4

G1 G2

L(G1)∈R1

G3

L(G3)∈R3

G4

L(G4)∈R4

L(G2)∈R2

2

1

3

4

5

6

7

8

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Figure 6. Hyper-graphs with different dimensions.

L(G3) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 −1 −1 0 −1 0 0 0

−1 3 0 −1 0 −1 0 0

−1 0 3 −1 0 0 −1 0

0 −1 −1 3 0 0 0 −1

−1 0 0 0 3 −1 −1 0

0 −1 0 0 −1 3 0 −1

0 0 −1 0 −1 0 3 −1

0 0 0 −1 0 −1 −1 3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

8×8

⇒ L(G3) ∈ R3



The smallest condensed matrices in the third contour of the factorization tree consist of

CCC, CCD, CDC, CDD, DCC, DCD, DDC, DDD

The 2 × 2 constituting parts of L(G3) are

A0 =⎡⎣ 3 −1

−1 3

⎤⎦ , A1 =

⎡⎣−1 0

0 −1

⎤⎦ , A2 =

⎡⎣ 3 −1

−1 3

⎤⎦ , A3 =

⎡⎣0 0

0 0

⎤⎦

⇒[

�C0 = 3 + (−1) = 2

�D0 = 3 + (−1) = 4

[�C1 = − 1

�D1 = − 1

[�C2 = �C1

�D2 = �D1

[�C3 = 0

�D3 = 0

As an example, the core DCD is as follows:

DCD = �D0 + �D1 + �D2 + �D3 = (4) + (−1) − (−1) − (0) = 4

8. SPECIALLY STRUCTURED GRAPHS

A graph is called specially structured if the corresponding Laplacian is specially structured,i.e.

LK×K = [lij ] : Iij ∈ Rn (58)

The Laplacian matrix contains block matrices which are hyper-symmetric. This is possible onlyif the considered graph is obtained by symmetric combination of hyper-graphs. As an example,the two graphs G1 and G2 are hyper-graphs and G∗ is obtained as their combination, Figure 7.

In order to calculate the eigenvalues of L(G∗), it is sufficient to form its eigenmatrices:

L(G∗) =

⎡⎢⎢⎢⎢⎢⎣

2 −1 −1 0

−1 2 0 −1

−1 0 2 −1

0 −1 −1 2

⎤⎥⎥⎥⎥⎥⎦

4×4

L(G∗) =⎡⎣A(2×2) B(2×2)

C(2×2) D(2×2)

⎤⎦ : A, B, C, D ∈ R1

eig[L(G∗)] =

⎡⎢⎢⎢⎢⎢⎣

�1 =[

A+D2

+1

2

√(A−D)2+4BC

]=[

4 −2

−2 4

]→

(�1C = 4+(−2) = 2

�1D = 4−(−2) = 6

�2 =[

A+D2

−1

2

√(A−D)2+4BC

]=[

2 −1

−1 2

]→

(�2C = 2+(−1) = 1

�2D = 2−(−1) = 3

(59)



1

2 2

1 3

4

3

4

Figure 7. Two hyper-graphs and their symmetric combination.

21

3 4

Figure 8. A graph with Form II symmetry.

At this stage one may ask why Form II symmetry is not used in the previous example, whichcould be achieved by appropriate re-numbering of the nodes, as shown in Figure 8.

If Form II symmetry is used, then the Laplacian will be as

L(G∗) =

⎡⎢⎢⎢⎢⎢⎣

2 −1 −1 0

0 4 0 −2

−1 0 2 −1

0 −2 0 4

⎤⎥⎥⎥⎥⎥⎦

4×4

The condensed matrices are then formed as

C =⎡⎣2 −1

0 4

⎤⎦+

⎡⎣−1 0

0 −2

⎤⎦ =

⎡⎣1 −1

0 2

⎤⎦ , D =

⎡⎣2 −1

0 4

⎤⎦−

⎡⎣−1 0

0 −2

⎤⎦ =

⎡⎣3 −1

0 6

⎤⎦

These two matrices have no more Form II symmetry and cannot be further refined. Thisproduced difficulties in the solution of large-scale problem.

RemarkA remarkable property of a specially structured matrix is that its eigendeterminant and eigen-matrices are hyper-symmetric.



ExampleCalculate the eigenvalues of the Laplacian matrix for the graph shown in Figure 9.

L(G∗) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4 −1 −1 0 −2 0 0 0−1 4 0 −1 0 −2 0 0

−1 0 4 −1 0 0 −2 00 −1 −1 4 0 0 0 −2

−2 0 0 0 7 −1 −1 −10 −2 0 0 −1 7 −1 −1

0 0 −2 0 −1 −1 7 −10 0 0 −2 −1 −1 −1 7

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

8×8

=[

A B

B C

]: A, B, C ∈ R2

Eigenmatrices are formed using

eig[L(G∗)] =

⎡⎢⎢⎢⎣

�1 =[

A + C2

+ 1

2

√(A − C)2 + 4B2

]

�2 =[

A + C2

− 1

2

√(A − C)2 + 4B2

]

�1 =

⎡⎢⎢⎢⎢⎢⎣

8.0322 −1 −1 −0.7962

−1 8.0322 −0.7962 −1

−1 −0.7962 8.0322 −1

−0.7962 −1 −1 8.0322

⎤⎥⎥⎥⎥⎥⎦

4×4

⇒

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

5.2361

9.2361

8.8284

8.8284

�2 =

⎡⎢⎢⎢⎢⎢⎣

2.9678 −1 −1 −0.2038

−1 2.9678 −0.2038 −1

−1 −0.2038 2.9678 −1

−0.2038 −1 −1 2.9678

⎤⎥⎥⎥⎥⎥⎦

4×4

⇒

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0.7639

3.1716

4.7639

3.1716

(60)

The eigenvalues of L(G∗) is obtained by

eig[L(G∗)] = {eig[�1]} ∪ {eig[�2]} (61)

One of the commonly used forms in the theory of graphs, is Form II symmetry with thefollowing form:

M =

⎡⎢⎢⎣

A B C

B A C

Ct Ct R

⎤⎥⎥⎦

3n×3n

(62)



2

1

3

4

5

6

7

8

Figure 9. A hyper-graph G∗.

If we want to consider this as a specially structured matrix, we make the following assumptionsand construct the form:

M =⎡⎢⎣

A B C

B A C

C C R

⎤⎥⎦

3n×3n

(63)

If A, B, C, and R ⊆ Ri , i.e. the blocks A, B and C are hyper-symmetric and Ct = C. Withthese assumptions M is a specially structured matrix and we can treat its blocks as entries ofa matrix to find the eigenvalues as

�1 = A − B

�2 = 12 (A + B + R) +

√(A + B − R)2 + 8C2

�3 = 12 (A + B + R) −

√(A + B − R)2 + 8C2

(64)

Example 1For the graph G shown in Figure 10, calculate the eigenvalue for the Laplacian matrix of G.

L(G) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

4 −1 −1 0 −1 −1−1 4 0 −1 −1 −1

−1 0 4 −1 −1 −10 −1 −1 4 −1 −1

−1 −1 −1 −1 4 0−1 −1 −1 −1 0 4

⎤⎥⎥⎥⎥⎥⎥⎥⎦

6×6

=

⎡⎢⎢⎣

A B C

B A C

Ct Ct R

⎤⎥⎥⎦

3n×3n

, eig[L(G)] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4

4

4

6

6

0



2

13

4

5

6

Figure 10. A graph G.

It can be observed that A, B, C and D are hyper-symmetric matrices. Therefore

�1 = A − B =⎡⎣ 5 −1

−1 5

⎤⎦

2×2

→ eig[�1] =[

64

�2 = 12 (A + B + R) +

√(A + B − R)2 + 8C2

�2 =[

5 1

1 5

]→ eig[�2] =

[6

4, �3 =

[2 −2

−2 2

]→ eig[�3] =

[4

0

eig[L(G)] = eig[�1] ∪ eig[�2] ∪ eig[�3] = {4, 4, 6, 6, 0}

which is the same result obtained for the entire matrix by Matlab software.

Example 2Consider the graph illustrated in Figure 11, and construct its Laplacian as

L(G) =

⎡⎢⎢⎢⎢⎣

3 −1 −1 −1

−1 3 −1 −1

−1 −1 3 −1

−1 −1 −1 3

⎤⎥⎥⎥⎥⎦

4×4 = 22 × 22

, L(G) ∈ R2

This matrix has a hyper-symmetry.The factorization tree is constructed as shown in Figure 12.As it was mentioned previously, this tree in the second contour results in 1 × 1 matrices. As

it has been shown before for

[L(G)] =⎡⎣A0 A1

A1 A0

⎤⎦ , eig[A0] = {�C0, �D0} and eig[A1] = {�C1, �D1} (65)



2

1

3

4

Figure 11. A hypergraph G.

L(G)22×22

C D

CC CD DC DD

contour 1

contour 2

Figure 12. The factorization tree.

and

eig[L(G)] = eig[CC] ∪ eig[CD] ∪ eig[DD] ∪ eig[DC] (66)

The constituting 2 × 2 submatrices of L(G) are

eig[CC] = �C0 + �C1, eig[DC] = �C0 + �C1

eig[CD] = �D0 + �D1, eig[DD] = �D0 + �D1

[A0] =⎡⎣ 3 −1

−1 3

⎤⎦ ⇒ eig[A0] =

[�C0 = 3 + (−1) = 2

�D0 = 3 − (−1) = 4

[A1] =⎡⎣−1 −1

−1 −1

⎤⎦ ⇒ eig[A1] =

[�C1 = − 1 + (−1) = − 2

�D1 = − 1 − (+1) = 0

eig[CC] = 2 + (−2) = 0, eig[CD] = 4 − 0 = 4

eig[DC] = 2 − (−2) = 4, eig[DD] = 4 − 0 = 4

eig[L(G)] = {0, 4, 4, 4}



1

5

6

2

3

4

Figure 13. The graph G.

15

6

2

3 4

Figure 14. Three-dimensional representation of G.

Example 3Consider the graph shown in Figure 13. The Laplacian matrix of G is a specially structuredmatrix, since its constituting blocks are hyper-symmetric (Figure 14).

L(G) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 0 −1 0 −1 −10 3 0 −1 −1 −1

−1 0 3 0 −1 −10 −1 0 3 −1 −1

−1 −1 −1 −1 5 −1−1 −1 −1 −1 −1 5

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

6×6

=

⎡⎢⎢⎣

A B C

B A C

Ct Ct R

⎤⎥⎥⎦

6×6

Therefore, the eigenvalue matrices are formed as

�1 =[

4 0

0 4

]→ eig[�1] = {4, 4}, �2 =

[6 0

0 6

]→ eig[�2] = {6, 6}

�3 =[

1 −1

−1 1

]→ eig[�3] = {0, 2}

The above figure can be considered as a space graph which can be taken as the units of aspace structure.



2

3 4

1

5 6

7 8

Figure 15. A hyper-symmetric graph G.

Example 4The graph shown in Figure 15 is a combined form of hyper-graph symmetric graphs.

The Laplacian matrix L(G) of G is a specially structured matrix.

L(G) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 0 −1 0 −1 0 −1 00 3 0 −1 0 −1 0 −1

−1 0 3 0 −1 0 −1 00 −1 0 3 0 −1 0 −1

−1 0 −1 0 4 −1 −1 00 −1 0 −1 −1 4 0 −1

−1 0 −1 0 −1 0 4 −10 −1 0 −1 0 −1 −1 4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

8×8

=

⎡⎢⎢⎢⎢⎣

A B B B

B A B B

B B C B

B B B C

⎤⎥⎥⎥⎥⎦

8×8

The eigenmatrices are constructed and their eigenvalues are calculated.

�1 = A − B =[

4 00 4

]→ eig[�1] = {4, 4}

�2 = C − B =[

5 −1−1 5

]→ eig[�2] = {6, 4}

�3 = B + 12 (A + C) + 1

2

√(A − C)2 + 16B2 =

[4.6180 −0.6180

−0.6180 4.6180

]

→ eig[�3] = {5.2361, 4}

�4 = B + 12 (A + C) − 1

2

√(A − C)2 + 16B2 =

[0.3820 −0.3820

−0.3820 0.3820

]

→ eig[�4] = {0.7639, 0}

eig[L(G)] =3⋃

i=1{�i} ⇒ eig[L(G)] = {4, 4, 4, 4, 5.2361, 6, 0.7639, 0}



RemarkDifferent combinations of hyper-symmetric graph, results in a specially structured graph. Theeigenvalues and determinant for the Laplacian matrix of this graph can be obtained usingalgebraic–matrix relationships.

9. HYPER-SYMMETRIC GRAPHS OF FORM III

In the previous sections, hyper-symmetric graphs of Form II were studied. Here, hyper-symmetric graphs of Form III are introduced. A graph G1 is called hyper-symmetric if fromeach node or edge of G1 an axis of symmetry passes. A hyper-symmetric graph of Form II isshown in Figure 16(a), which contains an even number of nodes.

A different type of hyper-symmetry which corresponds to a Form III symmetry is illustratedin Figure 16(b). In this graph, G2, an axis of symmetry passes from each node or edge andtherefore it is a hyper-symmetric graph, having an odd number of nodes.

Now consider the following matrix with Form III symmetry:

M =⎡⎢⎣

a b b

b a b

b b a

⎤⎥⎦

3×3

, a, b, c ∈ R (67)

(R3, +, ×) → ring (68)

R3 contains the set of matrices with the above-shown structure. This set together with twobinary operations + and × forms a commutative ring.

9.1. Properties

These matrices have the following properties:

(a) These matrices are commutative with respect to × , i.e.

M1 =⎡⎢⎣

a1 b1 b1

b1 a1 b1

b1 b1 a1

⎤⎥⎦ and M2 =

⎡⎢⎣

a2 b2 b2

b2 a2 b2

b2 b2 a2

⎤⎥⎦ (69)

M1 × M2 = M2 × M1 (70)

(b) The eigenvalues of such matrices are as follows:

M =⎡⎢⎣

a b b

b a b

b b a

⎤⎥⎦ ⇒ eig[M] =

⎡⎢⎣

�D = a − b

�D = a − b

�E = a + 2b

(71)



1

2

3

4 3

1

2

(a) G1 (b) G2

Figure 16. Two hyper-symmetric graphs: (a) G1; and (b) G2.

Hyper-symmetric matrices of Form II with higher dimensions can be defined as

R9 =

⎧⎪⎨⎪⎩M =

⎡⎢⎣

a b b

b a b

b b a

⎤⎥⎦ | a, b ∈ R3

⎫⎪⎬⎪⎭ (72)

The set R9 contains elements which are also hyper-symmetric. As an example, M9×9 has thefollowing structure:

R3i =

⎧⎪⎨⎪⎩M3i×3i =

⎡⎢⎣

a b b

b a b

b b a

⎤⎥⎦ | a, b ∈ R3i−1

⎫⎪⎬⎪⎭ (73)

M9×9 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a b b a′ b′ b′ a′ b′ b′b a b b′ a′ b′ b′ a′ b′b b a b′ b′ a′ b′ b′ a′

a′ b′ b′ a b b a′ b′ b′

b′ a′ b′ b a b b′ a′ b′b′ b′ a′ b b a b′ b′ a′

a′ b′ b′ a′ b′ b′ a b b

b′ a′ b′ b′ a′ b′ b a b

b′ b′ a′ b′ b′ a′ b b a

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

9×9

(74)



1

2

3

4

56

7

8

9

Figure 17. The complete graph K9.

The graph corresponding to M9×9 is a complete graph K9 as shown in Figure 17. In this case,the graph is also called a hyper-symmetric graph.

L(K9) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

8 −1 −1 −1 −1 −1 −1 −1 −1−1 8 −1 −1 −1 −1 −1 −1 −1−1 −1 8 −1 −1 −1 −1 −1 −1

−1 −1 −1 8 −1 −1 −1 −1 −1−1 −1 −1 −1 8 −1 −1 −1 −1−1 −1 −1 −1 −1 8 −1 −1 −1

−1 −1 −1 −1 −1 −1 8 −1 −1−1 −1 −1 −1 −1 −1 −1 8 −1−1 −1 −1 −1 −1 −1 −1 −1 8

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

9×9

The above matrix contains the following submatrices:

A =⎡⎢⎣

8 −1 −1

−1 8 −1

−1 −1 8

⎤⎥⎦ and B =

⎡⎢⎣

−1 −1 −1

−1 −1 −1

−1 −1 −1

⎤⎥⎦

9.2. Eigenvalues of specially structured matrices of high dimensions

As previously mentioned, if M is a 3 × 3 matrix, then we have a factorization tree as illustratedin Figure 18.

In this tree M3i×3i we have i contours and in the ith contour 1 × 1 matrices are created.As an example, for i = 2, M9×9 in the second contour 1 × 1 matrices are obtained. In suchmatrices, the eigenvalues are linear combinations of the eigenvalues of its constituting 3 × 3block matrices.



M3i×3i

D D E contour 1

contour 2

DD DD DD DD EEEDDE DE ED

contour i

Figure 18. The factorization tree M3i×3i .

As an example, for M9×9, if we have:

M9×9 =⎡⎢⎣

A B B

B A B

B B A

⎤⎥⎦

9×9

and A =⎡⎢⎣

a1 b1 b1

b1 a1 b1

b1 b1 a1

⎤⎥⎦

3×3

and B =⎡⎢⎣

a2 b2 b2

b2 a2 b2

b2 b2 a2

⎤⎥⎦

3×3

(75)

and

eig[A] =⎡⎢⎣

�D1

�D1

�E1

and eig[B] =⎡⎢⎣

�D2

�D2

�E2

(76)

then

eig[M]9×9 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�DD = �D1 − �D2

...

�DE = �E1 − �E2

...

�ED = �D1 + 2�D2

...

�EE = �E1 + 2�E2

(77)

Here, the first index from right to left for �, specifies the type of � and the second indexindicates the type of the operation being performed on the block matrices.



1 2

34

5

6

Figure 19. A space graph G.

ExampleThe eigenvalues for the Laplacian matrix of K9 is calculated as follows:

A =⎡⎢⎣

8 −1 −1

−1 8 −1

−1 −1 8

⎤⎥⎦ and B =

⎡⎢⎣

−1 −1 −1

−1 −1 −1

−1 −1 −1

⎤⎥⎦

⎡⎢⎣

�DA = 9

�DA = 9

�EA = 6

and

⎡⎢⎣

�DB = 0

�DB = 0

�EB = 3

According to Equation (77), the eigenvalues are calculated as

�DD = �DA − �DB = 9 − 0 = 9, �DD = 9, �DD = 9

�DD = 9, �DE = �EA − �EB = 6 + 3 = 9, �DE = 9

�ED = �DA + 2�DB = 9 + 2 × 0 = 9, �ED = 9

�EE = �EA − �EB = 6 − 2 × 3 = 0

eig[L(K9)] = {9, 9, 9, 9, 9, 9, 9, 9, 0}Therefore, the eigenvalues of the hyper-graph matrices are a combination of the eigenvalues ofthe constituting block matrices.

The following matrix is a specially structured matrix, since its blocks contain hyper-symmetricmatrices of Form III (Figure 19).

L(G) =

⎡⎢⎢⎢⎢⎢⎢⎣

3 −1 −1 −1 0 0−1 3 −1 0 −1 0−1 −1 3 0 0 −1

−1 0 0 3 −1 −10 −1 0 −1 3 −10 0 −1 −1 −1 3

⎤⎥⎥⎥⎥⎥⎥⎦

6×6



C =⎡⎣ 3 −1 −1

−1 3 −1−1 −1 3

⎤⎦+

⎡⎣−1 0 0

0 −1 00 0 −1

⎤⎦ =

⎡⎣ 2 −1 −1

−1 2 −1−1 −1 2

⎤⎦

D =⎡⎣ 3 −1 −1

−1 3 −1−1 −1 3

⎤⎦−

⎡⎣−1 0 0

0 −1 00 0 −1

⎤⎦ =

⎡⎣ 4 −1 −1

−1 4 −1−1 −1 4

⎤⎦

C and D are condensed matrices which are themselves hyper-symmetric.

�CD = 2 + (1) = 3, �CD = 2 + (1) = 3, �CE = 2 + (1) = 0

�DD = 4 − (−1) = 5, �CD = 4 − (−1) = 5, �CE = 4 + 2(−1) = 2

eig[L(G)] = {0, 2, 3, 3, 5, 5}

10. APPLICATIONS IN STRUCTURAL MECHANICS

In this section, examples of mass–spring systems are studied. It is shown how the presentmethods simplify the evaluation of their natural frequencies.

Example 1Consider the mass–spring system shown in Figure 20.

For k1 = k2 = k3 = k, the stiffness and mass matrices are constructed as

KS =

A B C D

A

B

C

D

⎡⎢⎢⎢⎢⎣

2k 0 −k 0

0 2k 0 −k

−k 0 2k −k

0 −k −k 2k

⎤⎥⎥⎥⎥⎦

and M =

⎡⎢⎢⎢⎢⎣

m 0 0 0

0 m 0 0

0 0 m 0

0 0 0 m

⎤⎥⎥⎥⎥⎦ (78)

It can be observed that both matrices are hyper-symmetric, and KS and M ∈ R2. Thus thematrix KS − �2M is also hyper-symmetric. Considering �2m = k, this matrix turns out tocorrespond to the characteristic equation of the system under study, i.e.

det[KS − �2M] = 0 (79)

Using the properties of the hyper-symmetric matrices, the eigenvalues of KS can easily beobtained. The decomposition tree is illustrated in Figure 21.

Equations (50)–(52), results in

A0 =[

2k 00 2k

]and A1 =

[−k 00 −k

]

CC = 2k + (−k) = k, CD = 2k + (−k) = k



k1 k2 k3 k2 k1

m m

CA

m

D B

m

Figure 20. A mass–spring system.

DDDCCD

DC

CCR2

R1

KS

Figure 21. The decomposition tree.

DC = 2k − (−k) = 3k, DD = 2k − (−k) = 3k

For �CC = k and �CD = k, �2m = k ⇒ � =√k/m

For �CC = 3k and �CD = 3k, �2m = 3k ⇒ � =√3k/m

In the previous example, if the stiffness of the springs are not selected identical, then

KS =

A B C D

A

B

C

D

⎡⎢⎢⎢⎢⎢⎣

k1 + k2 0 −k2 0

0 k1 + k2 0 −k2

−k2 0 k2 + k3 −k3

0 −k2 −k3 k2 + k3

⎤⎥⎥⎥⎥⎥⎦

Here KS becomes a matrix with special structure as

KS =[

A B

B C

]; A, B and C ∈ R1 ⇒

⎧⎨⎩

�1 = 12 (A + C) + 1

2

√(A − C)2 + 4B2

�2 = 12 (A + C) − 1

2

√(A − C)2 + 4B2

(80)

Now �1 and �2 are the eigenmatrices of the stiffness matrix of the system. Assuming theconsistency of the dimensions, and letting k1 = 1, k2 = 2 and k3 = 3, results in:

�1 =⎡⎣6.6316 −2.07

−2.07 6.6316

⎤⎦ ⇒ Eig(�1) =

{6.6316 − (−2.07) = 8.7016

6.6316 + (−2.07) = 4.5616

�2 =⎡⎣1.3684 −0.93

−0.93 1.3684

⎤⎦ ⇒ Eig(�2) =

{1.3684 − (−0.93) = 2.2984

1.3684 + (−0.93) = 0.4384



m m

BA D

m m

C

kS

k2 k1 k3 k4 k3k1

kS

Figure 22. Two systems connected by two springs.

The frequency can now be easily obtained as

�1 =√�i/m ⇒ �1 =√8.7016/m, �2 = √

4.5616/m

�3 =√2.2984/m, �4 =√0.4384/m

Example 2In this example, two systems are connected by means of two springs of stiffness kS , asillustrated in Figure 22.

The stiffness matrix is constructed in the following and the mass matrix is the same as thatof the previous example.

KS =

A B C D

A

B

C

D

⎡⎢⎢⎢⎢⎢⎣

k1 + k2 + kS −k2 −kS 0

−k2 k1 + k2 + kS 0 −kS

−kS 0 k3 + k4 + kS −k4

0 −kS −k4 k3 + k4 + kS

⎤⎥⎥⎥⎥⎥⎦

Here KS becomes a matrix with special structure of the following form:

KS =⎡⎣A B

B C

⎤⎦ ; A, B and C ∈ R1

Similar to the previous example, the eigenvalues and thus the frequencies of the system canbe obtained.

The blocks forming KS are as follows:

A =⎡⎣k1 + k2 + kS −k2

−k2 k1 + k2 + kS

⎤⎦ , B =

⎡⎣−kS 0

0 −kS

⎤⎦



C =⎡⎣k3 + k4 + kS −k4

−k4 k3 + k4 + kS

⎤⎦

It can be seen that these blocks are themselves hyper-matrices. Therefore, KS has the specialstructure and its eigenvalues can easily be calculated using Equation (80). For the numericalvalues as k1 = k2 = 1, k3 = k4 = 2 and kS = 3, we have

A =⎡⎣ 5 −1

−1 5

⎤⎦ , B =

⎡⎣−3 0

0 −3

⎤⎦ and C =

⎡⎣ 7 −2

−2 7

⎤⎦

Since �1 and �2 have hyper-symmetry, therefore the decomposition tree will have only onecontour, and the eigenvalues can be obtained from 1 × 1 matrices:

Eig(�1) ={

C1 = [9.1977 + (−1.6565)] = 7.5413

D1 = [9.1977 − (−1.6565)] = 10.8541

Eig(�2) ={

C2 = [2.8023 + (−1.3936)] = 1.4586

D2 = [2.8023 − (−1.3936)] = 4.1459

leading to Eig(KS) = {7.5413, 10.8541, 1.4586, 4.1459}. The frequencies of the system can theneasily be obtained by �i =√�i/m, �i ∈ {Eig(KS)}.

11. CONCLUDING REMARKS

Generalization of Form II and Form III symmetry enables a more vast application of canonicalforms in eigenproblems, such as studying the behaviour of vibrating systems, stability analysisand vibration of frames.

Hyper-symmetry makes more refined factorization of matrices and graphs feasible. The newclass of symmetry can result in a new graph product, the eigensolution of which can beobtained by the properties of its generators.

The applications of the methods developed in this paper are by no means limited to structuralengineering, and can be employed in other branches of science and engineering, whenevereigenvalues and eigenvectors are involved.

REFERENCES

1. Kaveh A, Sayarinejad MA. Eigensolutions for matrices of special patterns. Communications in NumericalMethods in Engineering 2003; 19:125–136.

2. Kaveh A, Sayarinejad MA. Eigensolutions for factorable matrices of special patterns. Communications inNumerical Methods in Engineering 2004; 20:133–146.

3. Kaveh A. Structural Mechanics: Graph and Matrix Methods (3rd edn). Research Studies Press: Somerset,U.K., 2004.

4. Kaveh A, Syarinejad MA. Eigenvalues of factorable matrices with Form IV symmetry. Communications inNumerical Methods in Engineering 2005; 21(6):269–278.



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Structures Technology, Topping BHV, Bittnar Z (eds). Saxe-Coburg Publication: Edinburgh, U.K., 2002(Chapter 12).

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NJ, 1976.13. Demmel JW. Applied Numerical Linear Algebra. SIAM: Philadelphia, PA, 1997.14. Kaveh A, Sayarinejad MA. Graph symmetry in dynamic systems. Computers and Structures 2004;

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eigensolution of specially structured matrices with hyper-symmetry

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