Study on Some Properties of Anti-centrosymmetric Matrices

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

[Vol-2, Issue-5 May- 2016]

Study on Some Properties of Anti-centrosymmetric Matrices Wenhui Lan1, Junqing Wang2 School of Science, Tianjin Polytechnic University, Tianjin, 300387, China

Abstract— In this paper, the anti-centrosymmetric matrices have been researched. According to the structural characteristics of the anti-centrosymmetric matrix, some new methods have been used to prove the necessary and sufficient conditions of a matrix being anti-centrosymmetric and its properties of eigenvalue and eigenvector; the nonsingularity of the anti-centrosymmetric matrices have been discussed,that the odd order anti-centrosymmetric matrix is singular has been obtained, and two methods of computing inverse of the matrices(even order) have been given.

Keywords— Anti-centrosymmetric Matrix, Centrosymmetric Matrix, Eigenvalue, Inverse Matrix.

I.

INTRODUCTION

Centrosymmetric and anti-centrosymmetric matrices are two kinds of important special matrices, which are widely used in the fields of information theory, numerical analysis, linear system theory and so on. At present, some significant achievements have been acquired from the research on the structure and properties, eigenvalues and inverse of centrosymmetric and anti-centrosymmetric matrices. For example, in reference[3], Tan Ruimei used the definition of anticentrosymmetric matrix to prove some new conclusions on adjoint matrix, eigenvalue and eigenvector of it; in reference[6], Liu Lianfu discussed the method of computing inverse of anticentrosymmetric matrices in the light of the structure and representation of it. In this paper, based on the previous literatures, some systematic summary research has been made and some new proof methods have been shown for anti-centrosymmetric matrices.

Definition 1. If A  ( aij )  C

nn

Definition 2. If A  ( aij )  C

nn

 an1i,n1 j , i, j  1, 2,, n ,then A is called centrosymmetric matrix.

For centrosymmetric matrix, the elements in line r from the end are precisely those in line r from the beginning which are nn arranged in the reverse order. The set of the whole n-order centrosymmetric matrices is denoted as CSR . and it meets aij

 an1i ,n1 j , i, j  1, 2,, n ,then A is called anti-centrosymmetric matrix. For anti-centrosymmetric matrix, the elements in line r from the end are precisely the opposite number of those in line r from the beginning which are arranged in the reverse order. The set of the whole n-order anti-centrosymmetric matrices nn is denoted as ACSR . and it meets aij

1    1    (en , , e1 ) is called n-order permutation matrix. Obviously, Definition 3. The matrix J n       1  Jn Jn  In .

 a11 a12  a21 a22 Definition 4. If A        am1 am 2

 a1n   amn   an 1,n  a2 n  ˆ , then A           amn   a1n

am,n 1 an 1,n 1  a1,n 1

 am1    an 1,1       a11 

is denoted as the inversion of matrix A .

Lemma 1.

A  ACSRnn if and only if A   J n AJ n .

Lemma2. If A  ACSR

2 n2 n

,then A can be expressed as Page | 15

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

 A A 1   J n A2 and if A  ACSR

(2 n1)(2 n1)

[Vol-2, Issue-5 May- 2016]

A2 J n    J n A1 J n 

, then A can be expressed as

 A1  A=   T  J A  n 2

 0  J n

A2 J n    T J n   J n A1 J n 

among which, A1 , A2 are n-order square matrices; J n is n-order permutation matrix;  ,  are n dimensional column vectors.

Lemma 3. A  ACSR

nn

Lemma 4. A  ACSR

nn

ˆ ; A  CSR if and only if A   A

nn

as well as A  ACSR

nn

if and only if A  O .

if and only if

ˆ ); (1)When n  2k , k  Z , A  ( A1 ,  A 1 *

ˆ ).

(2))When n  2k  1, k  Z , A  ( A1 ,  ,  A 1 *

among which,  is the column vector of A in line k  1 , A1 is constituted by the front k column elements of A with the

ˆ is the inversion of A . original order and A 1 1 Proof: When n  2k  1 , let A  ACSR

nn

 ( A1 ,  , Aˆ2 ) .From Lemma 3,

A  ( A1 ,  , Aˆ2 )   Aˆ  ( Aˆ2 , ˆ ,  Aˆ1 ) and from Definition 2 and Definition 4,

aij  a2 k 2i ,2 k 2 j and   ˆ , then

a11  a2 k 1,2 k 1 , , a1k   a2 k 1,k  2 ,

a1,k  2  a2 k 1,k , , a1n  a2 k 1,1

a21  a2 k ,2 k 1 ,  , a2 k  a2 k ,k  2 ,

a2,k  2  a2 k ,k ,  , a2 n  a2 k ,1

a2 k ,1  a2 n ,

 ，a2 k ,k =  a2,k  2 ,

a2 k 1,1  a1n ,  , a2 k 1,k  a1,k  2

a2 k ,k  2

  a2 k ,  , a2 k ,2 k 1  a21

a2 k 1,k  2  a1k , , a2 k 1,2 k 1  a11

ˆ , therefore, A  ( A ,  ,  Aˆ ) .

That is A1   A 2 1 1 Similarly, the conclusion(1) can be proved.

II. Theorem 1. If A  ACSR

nn

,

MAIN CONCLUSIONS

 is the eigenvector of A and 0 is the corresponding eigenvalue, then 0 is also one of the

eigenvalues which belong to A , besides,

J n is the eigenvector which is corresponding to 0 .

Proof: From what is known,

0 I  A  0 Because of Lemma 1,

J n (0 I ) J n  J n AJ n  0 Page | 16

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

[Vol-2, Issue-5 May- 2016]

and that is

J n 0 I  A J n  0 , 0 I  A  0 Then

(  ) 0 I  A  0 and 0 is the eigenvalue of A .

Since

A  0 , then  A  0 , and according to Lemma 1, J n AJ n  0

Premultiplying both sides of the equation yields

J n J n AJ n  (0 ) J n From Definition 3,

A( J n )  ( 0 ) J n which shows that J n is the eigenvector which is corresponding to 0 .

Theorem 2. If A, B  ACSR

nn

, then kA  ACSR nn , (k  C ) and A  B  ACSRnn , thus all of the n-order anti-

centrosymmetric matrices on the complex field constitute a linear subspace of C

n n

.

Theorem 3. The trace of a n-order anti-centrosymmetric matrix is 0, and therefore the sum of the matrix's eigenvalues is 0.

Proof: In terms of Definition 2,

aij  an1i ,n1 j , i, j  1, 2,, n For odd order anti-centrosymmetric matrices,the center of its elements must be 0. Then the sum of the elements on the diagonal is

a11  a22    ann  0 Because of all of the above,the trace of a n-order anti-centrosymmetric matrix is 0, and the sum of the eigenvalues is 0, too.

Theorem 4. If A is an n-order square matrix, then there exist the only B  CSR

nn

and C  ACSR

nn

, which can

indicate A  B  C .

Proof: Let

B then B  CSR

nn

and C  ACSR

nn

A  Aˆ A  Aˆ ,C  , 2 2

, the uniqueness will be proved in the following.

Suppose

A  B1  C1  B2  C2 among which

B1 , B2  CSR nn and C1 , C2  ACSRnn ,

then Page | 17

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

[Vol-2, Issue-5 May- 2016]

B1  B2  C1  C2 However,

B1  B2  CSR nn and C1  C2  ACSR nn , on the basis of Lemma 3,

B1  B2  C1  C2  O so

B1  B2 , C1  C2 Theorem 5. The product of a anti-centrosymmetric matrix and a same order centrosymmetric matrix is a anticentrosymmetric matrix; the product of two anti-centrosymmetric matrices of the same order is a centrosymmetric matrix.

Proof: (i)If A  ACSR

nn

and B  CSR

nn

,then

A   J n AJ n and B  J n BJ n hence

AB   J n AJ n J n BJ n   J n ( AB ) J n which means that AB  ACSR (ii）If A, B  ACSR

nn

nn

.

,then

A   J n AJ n and B   J n BJ n hence

AB  J n AJ n J n BJ n  J n ( AB) J n which means that

AB  CSRnn .

Theorem 6. Odd order anti-centrosymmetric matrices are singular.

Proof: Suppose A  ACSR

(2 n1)(2 n1)

and 1 , 2 , , n , n 1 is the n  1 eigenvalues of it.According to Theorem 3,

1  2    n  n 1  0 From Theorem 1, there are n pairs eigenvalues which is opposite with each other in 1 , 2 , , n , n 1 , and there must be a eigenvalue i

 0 left. Then 0 must be a eigenvalue of A, which implicates that det A  0. Consequently, odd order anti-

centrosymmetric matrices are singular.

In this paper, the nonsingular anti-centrosymmetric matrices can only be considered in the field of even order matrices.

Theorem 7. The converse of a nonsingular anti-centrosymmetric matrix is anti-centrosymmetric. 1

Proof: From what is given, A   J n AJ n and AA

 I n , then

J n AA1J n  J n AJ n J n A1J n  A(J n A1J n )  I n 1

1

Above with the uniqueness of A , A

  J n A1 J n can be deduced, and that is A1 is also anti-centrosymmetric.

Page | 18

International Journal of Engineering Research & Science (IJOER)

Theorem 8. If the nonsingular matrix A  ACSR

nn

ISSN: [2395-6992]

[Vol-2, Issue-5 May- 2016]

and B  ( A Ek ) , among which n  2k , k  Z and Ek is the *

first k columns of n-order identity matrix. Using elementary transformation to B , if A becomes identity matrix and 1

meanwhile Ek becomes matrix C , then A

 (C , Cˆ ) .

Proof: In view of Theorem 7, if A  ACSR

nn

1

and A is nonsingular, then A -1

which A1 is the matrix which is constituted by the first k columns of A , and

 ACSRnn . Let A1  ( A1 , A2 ) , among

A2 is the matrix which is constituted by the

1

end k columns of A . The same block-divided method can be used to I n , and let A( A1 , A2 )  ( Ek , G) .

Now,

( AA1 , AA2 )  ( Ek , G) so

AA1  Ek If A becomes I n by elementary transformation, then there exist elementary matrices P1 , P2 ,, Pm ,which can infer

PP 1 2  Pm A  I n .

Consequently, 1 A1  PP 1 2  Pm and A1  A Ek  PP 1 2  Pm Ek .

From all of the above, if elementary transformation is used to B  ( A Ek ) , and A 1

becomes the identity matrix, then Ek becomes the first k columns of A . Finally, according to Lemma 4(2), the conclusion can be obtained.

Theorem 9. If A  ACSR

2 n2 n



A1   J n A2

and that is A  

A2 J n   , then A is nonsingular if and only if  J n A1 J n 

B  A1  A2 and C  A1  A2 are all nonsingular, besides, 1 1 (C 1  B 1 ) J n  1 B C A    2   J n (C 1  B 1 )  J n ( B 1  C 1 ) J n  1

among which

A1 , A2 are n-order square matrices, and J n is n-order permutation matrix.

Proof: Let U



1  In  2  Jn

In   , then Jn   O C   A1 U T AU      B O    J n A2

and A   B C . Therefore,

A2 J n   A  J n AVJ 1 n

A is nonsingular if and only if B and C are nonsingular.

Moreover,

Page | 19

International Journal of Engineering Research & Science (IJOER)

 A1    J n A2

ISSN: [2395-6992]

[Vol-2, Issue-5 May- 2016]

A2 J n  1  B 1  C 1 (C 1  B 1 ) J n   I n     J n A1 J n  2   J n (C 1  B 1 )  J n ( B 1  C 1 ) J n   O

O  In 

so 1 1 (C 1  B 1 ) J n  1 B C A1    2   J n (C 1  B 1 )  J n ( B 1  C 1 ) J n 

 3 7 3  1    2 3 1 1 1  Example: If A  , then A  ?  1 1 3 2     1 3 7 3  Solve: 1. According to Theorem 8,

3 7  2 3 ( A E )    1 1  1 3 1 3  0 1  0 3   0 2

3 1 1 1 3 2 7 3 7 3 5  5 2 15 7 18 8

   

1 0 0 0    

0 0 0 1

0  1   1   1  0  2   0  3 0 1   0 0  1 0   0 0

3 7 3 1 3 2 3 1 1 7 3 1 0 1 0 0

0 0 1 0

0 0 0 1

0  0  1  0  1 3    5 8 5 4  (I , C)  18 3 4    0 2     

0 0 0 1

3 2 0   1   5 8  5 4  3 4 1 8  1 ˆ  then A  (C , C )  1 8 3 4 5 4 5 8    2 3 1   0 2. According to Theorem 9,

3 7  1 3   3 1 A1   , A2    , A2 J      2 3 1 1 1 1 then

 2 4  4 10  A1  A2     B , A1  A2   C  3 4 1 2  and

1  1  1 5   1 B1   ,C     3 4 1 2  1 2 2  so

Page | 20

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

[Vol-2, Issue-5 May- 2016]

3 2 0   1   (C  B ) J   5 8  5 4  3 4 1 8  1 B C A1     5 4 5 8  2   J (C 1  B 1 )  J ( B 1  C 1 ) J   1 8 3 4   2 3 1   0 1

1

1

1

REFERENCES [1] Xianfeng Wu, Xiaolin Zhang. Some Properties of Centre Skew Symmetric Matrix[J]. Journal of Science of Teachers' College and University: 2011,31（6）：3-5 [2] Fang Chen. A Simple Method of the Calculation of Centro-symmetric Matrices and Anti Centro-symmetric Matrices Inversion in Engineering[J]. Journal of Sichuan University of Science & Engineering(Natural Science Edition): 2006,19（1）：36-38 [3] Ruimei Tan, Xueli Xue. Probe on Some Properties of Centroskewsymmetric Matrices[J]. College Mathematics: 2010, 26（4）：173175 [4] Yongan Zhou, Tao Qin. Some Properties on Centrosymmetric Matrices[J]. Mathematics in Practice and Theory: 2007, 37（8）：186-188 [5] Bowei Tang. Reverse Order Law of Generalized Inverse and the Properties of Centrosymmetric Matrix[D]. Master Degree Thesis of Shanxi Normal University:2013,23-26 [6] Lianfu Liu, Changbo Zheng. Reversibility of Center Symmetric Matrices[J]. Journal of Shenyang Normal University(Natural Science): 2012,30（1）：23-26

Page | 21