International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 7, Issue 8, 2019, PP 3-7

ISSN No. (Print) 2347-307X & ISSN No. (Online) 2347-3142 DOI: http://dx.doi.org/10.20431/2347-3142.0708002 www.arcjournals.org

The Inequalities of Positive Semi-Definite Block with Partial Order Relations Hui Quan* Department of , Xiangtan University, Xiangtan411105, China

*Corresponding Author: Hui Quan, Department of mathematics, Xiangtan University, Xiangtan 411105, China

Abstract: In 2019，Zübeyde Ulukök obtained an important theorem in reference [1]：When ACis a H  * CB A positive semi-, then rr1  ,where A , B are n-order square matrices. In HAB3[11 ( ) ( )]  B this paper, we firstly do the same thing for a 33 positive semi-definite , and give a generalization of the above theorem. Next, we further generalize the case of kk positive semi-definite block matrix, and discuss the partial ordering relationship between the sum of matrices on quasi-diagonal lines and block matrices at other locations. Thus, we gave a new eigenvalue inequality. Keywords: Positive Semi-Definite Matrix Partial Order Relation

1. INTRODUCTION AND PRELIMINARIES Inequalities of positive semi-definite block matrices have been widely used in matrix theory. In recent years, inequalities about block matrices have become a hot topic of research. At the same time, some very good results have been obtained, such as references [1, 3, 4, 5]. In this paper, we mainly discussed some positive semi-definite block matrices and obtained some matrix inequalities. As we all know, the positive semi-definite block matrices have very good properties. Their eigenvalues are real numbers, so they can always be arranged in ascending order and we recorded then as

n ()()()AAA 21. In this paper, we use symbol 1()A to represent the largest eigenvalue of a positive semi-definite matrix A . And use symbol AB to represent BA be a positive semi-definite matrix, obviously, "" is a partial order relation. In particular, A  0denotes that matrix A is positive semi-definite. In addition, we call U a if it satisfiesUUI  ，we call A a Hermitian matrix if it satisfies AA  . Last, the AB denotes the direct sum of A and B , the block A 0 ; and 0 represents a zero block matrix.  0 B

2. MAIN RESULTS Let's start with the following lemmas

 Lemma1.1 Let AM mn, with mn , then (AA ) ( A A 0) with0Mmn .

Lemma1.2 Let be positive semi-definite matrix. Then , where denotes AM n n ()()AIAAI1 I

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the in . M n

AB Lemma1.3 If is a positive semi-definite matrix, then AB . BA

1 ABI Proof Just consider the equation ABII   0 and we can get the conclusion. 2 BAI 

AB Lemma1.4 If * is a positive semi-definite matrix, AC, are square matrices of the same BC order, then we have A C B B * and ACBB ()* .

AB Proof Becase* is a positive semi-definite matrix, So, for any unitary matrix U ,we have BC

* AB ABAB * UU*  0 , thus there is **UU0 . In particular, take BC BCBC

0 I ACBB* U  ,then we have .From lemma 1.3 we can get that  *  0 I 0 BBAC

. And we can draw another conclusion by replacing the original B with B .

Lemma1.5 If HM n is a positive semi-definite matrix, then for any nk unitary matrixU satisfying

* * UUI k and for eachik1,..., , we have i m k()()()H  i U HU  i H .

ADE Theorem 1 * is a positive semi-definite matrix, where A , B ,C are all n-order square HDBF  ** EFC matrices,then A rr1 for r 1. HABCB3[ ( )( )( )]  111 C ProofBecause H is a positive semi-definite matrix, so there exists an P such that

* HPP . We divide P into blocks: PXYZ[,,], where XYZM,,  3nn , and we can know that

XXA  , YYB  , ZZC  . From Lemma 1.1, we can get the following results:

*   111()(0)(XXX XA), 111()(0)(YYY YB ),

 1(ZZ )  1 ( Z Z  0)   1 ( C ).

Noting that the following equation holds: Hr()() P* P r P * PP * r 1 P ,and we denote

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T P P()*1r . So, for H r , there is the following decompositions：

* X*** TXX TYX TZXTTTX    r  ***    HP TPY TXY TYY TZYTTTY    *** Z TXZ TYZ TZZTTTZ   

*  11 TTT TT 220000 We notice that TTT , and .  11 TTTT  3() TTTTT   220000   TTT TTT 11  22 TT  0000   So, from lemma 1.2. We can get that: * XXAA HTYYTBPPBrr3().3()3()  *1 111   ZZCC

A A ***1 r  3() r1***XXYYZZB 3[()()()]XXYYZZB 1 111  C C

A . So, the theorem is proved. 3[()()()]ABCB r1  111 C On the basis of this conclusion, we can easily get the results of Zübeyde Ulukök in reference[1]:

AC Corollary 1  is a positive semi-definite matrix, where A , B are n-order square H  * CB matrices,then

rr1 A HAB3[()()]11 for r 1. B Next, we will discuss the case of kk positive semi-definite block matrices and explore the relationship between quasi-diagonal matrices and other block matrices at other locations.

MMM11121 k Theorem 2 MMM* is a positive semi-definite matrix, and M , …, M are H  12222 k 11 kk  ** MMM12kkkk all n-order square matrices, if it satisfied M ij are all Hermitian matrices, for

(1i  j  n ), then

k 2 MMii ij i11k 1  i  j  n

Proof First of all, notice the following facts: ifU1 ,…, U k are all unitary matrices, A is a positive

** semi-definite matrix, thenU11 AU... Ukk AU is still a positive semi-definite matrix.

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kk1 MM iiil ： After calculation, we can get that  0 . From lemma 1.4 we can get that ili11MMilll k11 k k k ,  (MMMMMii ll )    ( il  il )  2  ij i1 l  i  1 i  1 l  i  1 1  i  j  n k 2 So, there is MMiiij . iijn11k 1

AB Corollary 2If * is a positive semi-definite matrix, AC, are square matrices of the same BC order, and B is a Hermitian matrix, then trBtrAtrB max, .

MMM11121 k * Theorem3 MMM12222 k is a positive semi-definite matrix, and M , …, M are all H   11 kk  ** MMM12kkkk n-order square matrices, if it satisfied M ij are all Hermitian matrices, for(1 i j  n ),then for these

1in, we have 2 iiji()() MH k 1ijn  

I  * Proof Take a unitary matrix 1 I , thenU satisfiedUUI  n , noticed that M ij are all Hermitian U   k   I matrices, for ( ),so we have

k * 12 122 U HUMM iiij+ ,  MMijij+ (from Theorem 2) kkiij11 n   k kk111ij  nij  n 

2 1 2 MMij(  1)  ij . k11i  j  n k1 k  i  j  n

2  Then for 1in, from lemma 1.5 we have iijii()()( MU )  HUH , thus the k 1i  j  n

2 conclusion i()() MH ij i is proved. k 1i  j  n

REFERENCES [1] Zübeyde Ulukök, More inequalities for positive semidefinite 2 ×2block matrices and their blocks [J]. Linear Algebra and its Applications 572 (2019) 51–67 [2] X. Zhan, Matrix Inequalities, Lecture Notes in Math., vol. 1790, Springer-Verlag, Berlin, 2002.

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[3] Y. Zhang, Eigenvalue majorization inequalities for positive semidefinite block matrices and their blocks, Linear Algebra Appl. 446 (2014) 216–223. [4] F. Zhang, Matrix inequalities by means of block matrices, Mathematical Inequalities and Applications, Vol. 4, No. 4, (2001), 481-490. [5] R. Türkmen and Z. Ulukök, Inequalities for singular values of positive semidefinite block matrices[J]. International Mathematical Forum, Vol. 6, 2011, no. 31, 1535 – 1545

Citation: Hui Quan (2019).The Inequalities of Positive Semi-Definite Block Matrix with Partial Order Relations. International Journal of Scientific and Innovative Mathematical Research (IJSIMR), 7(8), pp. 3-7. http://dx.doi.org/ 10.20431/2347 -3142.0708002 Copyright: © 2019 Authors, this is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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