Trace inequalities for positive semidefinite matrices with centrosymmetric structure

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1 Zhao et al Journal of Inequalities pplications 1, 1:6 RESERCH Trace inequalities for positive semidefinite matrices with centrosymmetric structure Di Zhao 1, Hongyi Li 1* Zhiguo Gong Open ccess * Correspondence: Hongyili_buaa@163com 1 LMI, School of Mathematics System Science, eihang University, eijing, China ull list of author information is available at the end of the article bstract In this article, we present some results on the Hadamard product of positive semidefinite matrices with centrosymmetric structure ased on these results, several trace inequalities on positive semidefinite centrosymmetric matrices are obtained 1 Introduction preliminaries We will use the following notation Let C n n R n n be the space of n n complex real matrices, respectively The identity matrix in C n n is denoted by I I n Let T, Ā, H,tr( denote the transpose, the conjugate, the conjugate transpose, the trace of a matrix, respectively Let Re(a represent the real part of a The robenius inner product <, > in C m n over the complex field is defined as follows: <, > Re(tr( H, for, Î C m n, ie, <, > is the real part of the trace of H The induced matrix norm is <, > Re(tr( H tr( H, which is called the robenius (Euclidean norm matrixî C n n is Hermitian if H, is called positive semidefinite, written as (see [1, p 159], if x H x, x C n (1:1 is further called positive definite, symbolized >, if the strict inequality in (11 holds for all non-zero x Î C n n equivalent condition for Î C n to be positive definite is that is Hermitian all eigenvalues of are positive Let be two Hermitian matrices of the same size If - is positive semidefinite, we write Next, we introduce some basic definitions lemmas Definition 11 (see [1] Let be a square complex matrix partitioned as (1: where 11 is a square submatrix of If 11 is nonsingular, we call à the Schur complement of 11 in Note If is a positive definite matrix, then 11 is nonsingular à 11 1 Zhao et al; licensee Springer This is an Open ccess article distributed under the terms of the Creative Commons ttribution License ( which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited

2 Zhao et al Journal of Inequalities pplications 1, 1:6 Page of 6 Definition 1 (see [] (a ij Î C n n is called a centrosymmetric matrix, if a ij a n i+1,n j+1, 1 i n,1 j n, or J n J n, where J n (e n, e n-1,, e 1, e i denotes the unit vector with the ith entry 1 If a matrix is both positive semidefinite 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 []: Lemma 11 (see [] Let (a ij Î C n n (n be centrosymmetric Then, has the following form, ( Jm CJ m CJ m J m, P T P Jm C, (1:3 + J m C where C m m, C C m m, P 1 Im I m J m J m Note In this article, we mainly discuss the case n orn is odd, ie, n + 1, similar results can be obtained by taking similar steps Recently,in[3],Ulukök Türkmen proved some matrix trace inequalities for positive semidefinite matrices: Lemma 1 (see [3] Let, Î C n n Then, ( m ( H m ( H m (1:4 where m is an positive integer, sts for the Hadamard product of Note Particularly, if in Lemma 1 are both semidefinite matrices, then ( m Lemma 13 (see [3] Let i Î C n n,(i 1,,,k be semidefinite matrices Then, for positive real numbers s, m, t ( k ( k ( ((s+t/ k i sm i tm i (1:5 i1 i1 11 Lemma 14 (see [3] Let 1 11, 1 Then, 1 1 ] 1/ ] 1/ tr[ (Ã 11 + tr[ ( 1/ 1/ 11 tr( m tr( m m, (1:6 where m is an integer Main results Lemma 1 Let (a ij n n, (b ij n n (n be two centrosymmetric matrices with the following form: 1 J m J m, J m 1 J m ( 1 J m J m J m 1 J m Then, is a centrosymmetric matrix i1, 1,, 1, C m m (:1

3 Zhao et al Journal of Inequalities pplications 1, 1:6 Page 3 of 6 Proof y the definition of Hadamard product, 1 J m J m 1 J m J m 1 1 (J m J m (J m J m J m 1 J m J m 1 J m (J m 1 J m (J m 1 J m We shall prove the following { (Jm 1 J m (J m 1 J m J m ( 1 1 J m, (J m J m (J m J m J m ( 1 J m (: (:3 rom (1, a 11 a 1m b 11 b 1m a 11 b 11 a 1m b 1m 1 1, (:4 a m1 a mm b m1 b mm a m1 b m1 a mm b mm a mm b mm a m1 b m1 J m ( 1 1 J m (:5 a 1m b 1m a 11 b 11 Since a mm a m1 b mm b m1 J m 1 J m, J m 1 J m, (:6 a 1m a 11 b 1m b 11 we have the following a mm b mm a m1 b m1 (J m 1 J m (J m 1 J m (:7 a 1m b 1m a 11 b 11 rom (5 (7, it is clear that (J m 1 J m (J m 1 J m J m ( 1 1 J m (:8 Similarly, we can prove that (J m J m (J m J m J m ( J m (:9 rom (8 (9, we can see that (3 holds y Lemma 11, is a centrosymmetric matrix Lemma (see [4] Let (a ij n n (n a positive semidefinite centrosymmetric matrix with the following form Jm CJ m,, C C m m CJ m J m Let M - J m C N + J m C Then, M, N are positive semidefinite matrices Theorem 1 Let, Î C n n (n be two positive semidefinite centrosymmetric matrices with the same form as in (1 Let M 1 J m, N 1 + J m, M 1 J m, N 1 + J m,

4 Zhao et al Journal of Inequalities pplications 1, 1:6 Page 4 of 6 X 1 1 J m (, Y J m ( Then, the following inequality holds: X m + ( Y m M + N ( M + N (:1 Proof Since, are centrosymmetric matrices, by Lemma 1, is centrosymmetric 1 1 (J m J m (J m J m (J m 1 J m (J m 1 J m rom Lemma 11, P T ( P P is orthogonal, then ( P T ( m X P m Y m, 1 1 J m ( 1 1 J m ( ( m P T ( m P X m Y m X m + Y m Similarly from Lemma 11, ( P T 1 J P 1 M 1 + J 1 N ( P T 1 J P 1 M 1 + J 1 N Then, P T P ( M N, P T M P, N M N M N ( M + ( N M + N Since both are positive semidefinite matrices, by Lemma 1 Thus, ( X m + ( Y m M + N, ( M + N X Y

5 Zhao et al Journal of Inequalities pplications 1, 1:6 Page 5 of 6 Theorem Let Î C n n (n be postive semidefinite centroysymmetric with the form: Jm CJ m,, C C m m CJ m J m Let M - J m C, N + J m C Then, for positive integers s, m, t, the following two equalities hold ((s+t/ M ((s+t/ N + ((s+t/, (:11 M ((s+t/ N + ((s+t/ ( M sm + N sm ( M tm + N tm (:1 Proof y Lemma 11, P T Jm C P + J m C M N Since is positive semidefinite, we have P T ((s+t/ M ((s+t/ P N ((s+t/ Thus, ((s+t/ P T ((s+t/ M ((s+t/ N ((s+t/ M ((s+t/ N + ((s+t/ rom Lemma, M N are positive semidefinite Combining Lemma 13, we have M ((s+t/ N + ((s+t/ ( M sm + N sm ( M tm + N tm Theorem 3 Let, Î C n n (n are positive semidefinite centrosymmetric matrices with the same form as in (1 Let M 1 J m, N 1 + J m, M 1 J m, N 1 + J m Then, the following inequality holds tr M 1/ M1/ + tr N 1/ N1/ tr( m tr ( m m (:13 Proof rom Lemma 11, there exists an orthogonal matrix P such that P T 1 J P m M, 1 + J m N P T 1 J P m M 1 + J m N

6 Zhao et al Journal of Inequalities pplications 1, 1:6 Page 6 of 6 ccording to Definition 11, Ñ M N 1 rom Lemma 14, M, M N M 1 N tr M 1/ M1/ + tr N 1/ N1/ tr(p T PP T P m tr ( P T P m Since P is orthogonal, we have tr(p T P m tr( m y Lemma 14, the following holds tr M 1/ M1/ + tr N 1/ N1/ tr( m tr ( m m cknowledgements We would like to thank the reviewers for providing valuable comments suggestions to improve the manuscript This study was supported by the National Natural Science oundation of China (No uthor details 1 LMI, School of Mathematics System Science, eihang University, eijing, China aculty of Science Technology, University of Macau, C Postal 31 Macau, China uthors contributions DZ carried out studies on the linear algebra matrix theory with applications, drafted the manuscript HL helped prove some lemmas theorems ZG read the manuscript carefully gave valuable suggestions comments ll authors read approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 4 July 11 ccepted: 1 March 1 Published: 1 March 1 References 1 Zhang, : Matrix Theory: asic Results Techniques Springer, New York (1999 Liu, ZY: Some properties of centrosymmetric matrices ppl Math Comput 141, 17 6 ( 3 Ulukök, Z, Türkmen, R: On some matrix trace inequalities J Inequal ppl 1 (1 rticle ID Li, HY, Gao, ZS, Zhao, D: note on robenius conditional number with positive definite matrices J Inequal ppl11 doi:11186/19-4x-1-6 Cite this article as: Zhao et al: Trace inequalities for positive semidefinite matrices with centrosymmetric structure Journal of Inequalities pplications 1 1:6 Submit your manuscript to a journal benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropencom