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Trace inequalities for positive semidefinite matrices with centrosymmetric structure

Abstract

In this article, we present some results on the Hadamard product of positive semidefinite matrices with centrosymmetric structure. Based on these results, several trace inequalities on positive semidefinite centrosymmetric matrices are obtained.

1 Introduction and preliminaries

We will use the following notation. Let n × nand n × nbe the space of n × n complex and real matrices, respectively. The identity matrix in n × nis denoted by I = I n . Let AT, Ā, AH, and tr(A) denote the transpose, the conjugate, the conjugate transpose, and the trace of a matrix A, respectively. Let Re(a) represent the real part of a. The Frobenius inner product < ·, · > F in m × nover the complex field is defined as follows: < A, B > F = Re(tr(BHA)), for A, B m × n, i.e., < A, B > is the real part of the trace of BHA. The induced matrix norm is A F = < A , A > F = R e ( t r ( A H A ) ) = t r ( A H A ) , which is called the Frobenius (Euclidean) norm.

A matrix A n × nis Hermitian if AH= A, and A is called positive semidefinite, written as A ≥ 0 (see [1, p. 159]), if

x H A x 0 , x n .
(1.1)

A is further called positive definite, symbolized A > 0, if the strict inequality in (1.1) holds for all non-zero x n. An equivalent condition for A nto be positive definite is that A is Hermitian and all eigenvalues of A are positive.

Let A and B be two Hermitian matrices of the same size. If A - B is positive semidefinite, we write

A B .

Next, we introduce some basic definitions and lemmas.

Definition 1.1 (see [1]). Let A be a square complex matrix partitioned as

A = A 11 A 12 A 21 A 22
(1.2)

where A11is a square submatrix of A. If A11is nonsingular, we call à 11 = A 22 - A 21 A 11 - 1 A 12 the Schur complement of A11in A.

Note. If A is a positive definite matrix, then A11 is nonsingular and A22Ã11 ≥ 0.

Definition 1.2 (see [2]). A = (a ij ) n × nis called a centrosymmetric matrix, if

a i j = a n - i + 1 , n - j + 1 , 1 i n , 1 j n , o r J n A J n = A ,

where J n = (e n , en-1,..., e1), e i denotes the unit vector with the ith entry 1.

If a matrix is both positive semidefinite and centrosymmetric, we call this matrix positive semidefinite centrosymmetric.

Using the partition of matrix, the central symmetric character of a square centrosymmetric matrix can be described as follows [2]:

Lemma 1.1 (see [2]). Let A = (a ij ) n × n(n = 2m) be centrosymmetric. Then, A has the following form,

A = B J m C J m C J m B J m , and P T A P = B - J m C 0 0 B + J m C ,
(1.3)

where B m × m , C m × m , P = 1 2 I m I m - J m J m .

Note. In this article, we mainly discuss the case n = 2m. For n is odd, i.e., n = 2m + 1, similar results can be obtained by taking similar steps.

Recently, in [3], Ulukö k and Tü rkmen proved some matrix trace inequalities for positive semidefinite matrices:

Lemma 1.2 (see [3]). Let A, B n × n. Then,

( A B ) m F 2 A H A m F B H B m F
(1.4)

where m is an positive integer, A B stands for the Hadamard product of A and B.

Note. Particularly, if A and B in Lemma 1.2 are both semidefinite matrices, then A B m F 2 A 2 m F B 2 m F .

Lemma 1.3 (see [3]). Let A i n × n, (i = 1,2,...,k) be semidefinite matrices. Then, for positive real numbers s, m, t

i = 1 k A i ( ( s + t ) / 2 ) m F 2 i = 1 k A i s m F 2 i = 1 k A i t m F 2
(1.5)

Lemma 1.4 (see [3]). LetA= A 11 A 12 A 21 A 22 0,B= B 11 B 12 B 21 B 22 0. Then,

t r à 22 1 / 2 B 11 1 / 2 2 m + t r ( A 22 ) 1 / 2 B ̃ 11 1 / 2 2 m t r ( A B ) m t r ( A m B m ) ,
(1.6)

where m is an integer.

2 Main results

Lemma 2.1. Let A = (a ij )n × n, B = (b ij )n × n(n = 2m) be two centrosymmetric matrices with the following form:

A = A 1 J m A 2 J m A 2 J m A 1 J m , B = B 1 J m B 2 J m B 2 J m B 1 J m , A 1 , A 2 , B 1 , B 2 m × m
(2.1)

Then, A B is a centrosymmetric matrix.

Proof. By the definition of Hadamard product,

A B = A 1 J m A 2 J m A 2 J m A 1 J m B 1 J m B 2 J m B 2 J m B 1 J m = A 1 B 1 ( J m A 2 J m ) ( J m B 2 J m ) A 2 B 2 ( J m A 1 J m ) ( J m B 1 J m )
(2.2)

We shall prove the following

( J m A 1 J m ) ( J m B 1 J m ) = J m ( A 1 B 1 ) J m , ( J m A 2 J m ) ( J m B 2 J m ) = J m ( A 1 B 2 ) J m .
(2.3)

From (2.1),

A 1 B 1 = a 11 a 1 m a m 1 a m m b 11 b 1 m b m 1 b m m = a 11 b 11 a 1 m b 1 m a m 1 b m 1 a m m b m m ,
(2.4)

and

J m ( A 1 B 1 ) J m = a m m b m m a m 1 b m 1 a 1 m b 1 m a 11 b 11 .
(2.5)

Since

J m A 1 J m = a m m a m 1 a 1 m a 11 , and J m B 1 J m = b m m b m 1 b 1 m b 11 ,
(2.6)

we have the following

( J m A 1 J m ) ( J m B 1 J m ) = a m m b m m a m 1 b m 1 a 1 m b 1 m a 11 b 11 .
(2.7)

From (2.5) and (2.7), it is clear that

J m A 1 J m J m B 1 J m = J m A 1 B 1 J m .
(2.8)

Similarly, we can prove that

J m A 2 J m J m B 2 J m = J m A 2 B 2 J m .
(2.9)

From (2.8) and (2.9), we can see that (2.3) holds. By Lemma 1.1, A B is a centrosymmetric matrix.

Lemma 2.2 (see [4]) Let A = (a ij )n × n(n = 2m) a positive semidefinite centrosymmetric matrix with the following form

A = B J m C J m C J m B J m , B , C m × m .

Let M = B - J m C and N = B + J m C. Then, M, N are positive semidefinite matrices.

Theorem 2.1 Let A, B n × n(n = 2m) be two positive semidefinite centrosymmetric matrices with the same form as in (2.1). Let

M A = A 1 - J m A 2 , N A = A 1 + J m A 2 , M B = B 1 - J m B 2 , N B = B 1 + J m B 2 ,

and

X = A 1 B 1 - J m A 2 B 2 , Y = A 1 B 1 + J m A 2 B 2 .

Then, the following inequality holds:

X m F 2 + Y m F 2 M A 2 m F 2 + N A 2 m F 2 M B 2 m F 2 + N B 2 m F 2 .
(2.10)

Proof. Since A, B are centrosymmetric matrices, by Lemma 2.1, A B is centrosymmetric and

A B = A 1 B 1 ( J m A 2 J m ) ( J m B 2 J m ) A 2 B 2 ( J m A 1 J m ) ( J m B 1 J m ) .

From Lemma 1.1,

P T A B P = A 1 B 1 - J m A 2 B 2 0 0 A 1 B 1 - J m A 2 B 2 = X 0 0 Y .

P is orthogonal, then

P T A B m P = X m 0 0 Y m ,

and

A B m F 2 = P T A B m P F 2 = X m 0 0 Y m F 2 = X m F 2 + Y m F 2 .

Similarly from Lemma 1.1,

P T A P = A 1 - J 1 A 2 0 0 A 1 + J 1 A 2 = M A 0 0 N A , P T B P = B 1 - J 1 B 2 0 0 B 1 + J 1 B 2 = M B 0 0 N B .

Then,

P T A 2 m P = M A 2 m 0 0 N A 2 m , P T B 2 m P = M B 2 m 0 0 N B 2 m ,

and

A 2 m F B 2 m F = M A 2 m 0 0 N A 2 m F M B 2 m 0 0 N B 2 m F = M A 2 m F 2 + N A 2 m F 2 M B 2 m F 2 + N B 2 m F 2 .

Since both A and B are positive semidefinite matrices, by Lemma 1.2

A B F 2 A 2 m F B 2 m F .

Thus,

X m F 2 + Y m F 2 M A 2 m F 2 + N A 2 m F 2 M B 2 m F 2 + N B 2 m F 2 .

Theorem 2.2. Let A n × n(n = 2m) be postive semidefinite centroysymmetric with the form:

A = B J m C J m C J m B J m , B , C m × m .

Let M = B - J m C, N = B + J m C. Then, for positive integers s, m, t, the following two equalities hold

A ( ( s + t ) / 2 ) m F 2 = M ( ( s + t ) / 2 ) m F 2 + N ( ( s + t ) / 2 ) m F 2 ,
(2.11)

and

M ( ( s + t ) / 2 ) m F 2 + N ( ( s + t ) / 2 ) m F 2 M s m F 2 + N s m F 2 M t m F 2 + N t m F 2 .
(2.12)

Proof. By Lemma 1.1,

P T A P = B - J m C 0 0 B + J m C = M 0 0 N .

Since A is positive semidefinite, we have

P T A ( ( s + t ) / 2 ) m P = M ( ( s + t ) / 2 ) m 0 0 N ( ( s + t ) / 2 ) m .

Thus,

A ( ( s + t ) / 2 ) m F 2 = P T A ( ( s + t ) / 2 ) m F 2 = M ( ( s + t ) / 2 ) m 0 0 N ( ( s + t ) / 2 ) m F 2 = M ( ( s + t ) / 2 ) m F 2 + N ( ( s + t ) / 2 ) m F 2 .

From Lemma 2.2, M and N are positive semidefinite. Combining Lemma 1.3, we have

M ( ( s + t ) / 2 ) m F 2 + N ( ( s + t ) / 2 ) m F 2 M s m F 2 + N s m F 2 M t m F 2 + N t m F 2 .

Theorem 2.3 Let A, B n × n(n = 2m) are positive semidefinite centrosymmetric matrices with the same form as in (2.1). Let

M A = A 1 - J m A 2 , N A = A 1 + J m A 2 , M B = B 1 - J m B 2 , N B = B 1 + J m B 2 .

Then, the following inequality holds

t r M A 1 / 2 M B 1 / 2 2 m + t r N A 1 / 2 N B 1 / 2 2 m t r ( A B ) m t r A m B m .
(2.13)

Proof. From Lemma 1.1, there exists an orthogonal matrix P such that

P T A P = A 1 - J m A 2 0 0 A 1 + J m A 2 = M A 0 0 N A , P T B P = B 1 - J m B 2 0 0 B 1 + J m B 2 = M B 0 0 N B .

According to Definition 1.1,

Ñ A = M A - 0 N A - 1 0 = M A , M ̃ B = N B - 0 M B - 1 0 = N B .

From Lemma 1.4,

t r M A 1 / 2 M B 1 / 2 2 m + t r N A 1 / 2 N B 1 / 2 2 m t r ( P T A P P T B P ) m = t r P T A B P m .

Since P is orthogonal, we have tr(PTABP)m= tr(AB)m. By Lemma 1.4, the following holds

t r M A 1 / 2 M B 1 / 2 2 m + t r N A 1 / 2 N B 1 / 2 2 m t r ( A B ) m t r A m B m .

References

  1. Zhang F: Matrix Theory: Basic Results and Techniques. Springer, New York; 1999.

    Chapter  Google Scholar 

  2. Liu ZY: Some properties of centrosymmetric matrices. Appl Math Comput 2002, 141: 17–26.

    MathSciNet  Google Scholar 

  3. Ulukök Z, Türkmen R: On some matrix trace inequalities. J Inequal Appl 2010., 2010: Article ID 201486

    Google Scholar 

  4. Li HY, Gao ZS, Zhao D: A note on Frobenius conditional number with positive definite matrices. J Inequal Appl 2011:

    Google Scholar 

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Acknowledgements

We would like to thank the reviewers for providing valuable comments and suggestions to improve the manuscript. This study was supported by the National Natural Science Foundation of China (No. 60831001).

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Correspondence to Hongyi Li.

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The authors declare that they have no competing interests.

Authors' contributions

DZ carried out studies on the linear algebra and matrix theory with applications, and drafted the manuscript. HL helped prove some lemmas and theorems. ZG read the manuscript carefully and gave valuable suggestions and comments. All authors read and approved the final manuscript.

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Zhao, D., Li, H. & Gong, Z. Trace inequalities for positive semidefinite matrices with centrosymmetric structure. J Inequal Appl 2012, 62 (2012). https://doi.org/10.1186/1029-242X-2012-62

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Keywords

  • Positive Real Number
  • Orthogonal Matrix
  • Positive Semidefinite
  • Positive Definite Matrix
  • Real Matrice