Some Inequalities About Trace of Matrix

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Journal of Scientific and Engineering Research, 2019, 6(7):89-93

ISSN: 2394-2630 Research Article CODEN(USA): JSERBR

Some Inequalities about Trace of Matrix

TU Yuanyuan*1, SU Runqing2

1Department of Mathematics, Taizhou College, Nanjing Normal University, Taizhou 225300, China Email:[email protected] 2College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China Abstract In this paper, we studied the inequality of trace by using the properties of the block matrix, singular value and eigenvalue of the matrix. As a result, some new inequalities about trace of matrix under certain conditions are given, at the same time, we extend the corresponding results.

Keywords eigenvalue, singular value, trace, inequality 1. Introduction

For a complex number x  a  ib , where a,b are all real, we write Re x  a, Im x  b , as usual , let I n mn be an n n unit matrix, M be the set of m n complex matrices, diag d1,,L dn  be the diagonal matrix with diagonal elements dd1,,L n , if A be a matrix, denote the eigenvalues of by  i A, H the singular values of A by  i A, the trace of by trA , the associate matrix of A by A , A be a semi- positive definite matrix if be a Herimite matrix and i  A  0. In 1937 , Von Neumann gave the famous inequality [1] n (1.1) |tr ( AB ) | ii ( A ) ( B ), i1 where AB, are nn complex matrices. Scholars were very active in the study of this inequality, and obtained many achievements. In 2007, N. Komaroff extended the (1.1) by using singular value decomposition, and got the equality [2] n RetrVAWB   Rei T  i A i B. (1.2) i1 whereT  VPQWRS, Obviously, when V  B  I n , (1.2) yielded the equality n (1.3) RetrAW    Rei T  i A. i1 where T  PQW. Let A M mn , denote the singular value decomposition of A by

A  P1,OQ, (1.4)

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Yuanyuan TU & Runqing SU Journal of Scientific and Engineering Research, 2019, 6(7):89-93

where 1  diag1 A,, m A, 1 A    m A  0 , O is a mn  m null matrix, P,Q are unitary matrices of order m and n respectively. Let B  M nm , then denote the singular value decomposition of B by

 2  B  R S, (1.5)  O    diag B,, B,  B    B  0 where 2 1 m 1 m , R, S are unitary matrices of order and respectively. In fact, for the m n matrix A and the n m matrix B ,the equation trAB  trBA always hold, so we might as well assume mn in the singular value decomposition. This paper needs the following lemmas. Lemma 1 [3] Let x, y  Rn , then n n n  xi yi   xi yi   xi yi. i1 i1 i1 nn Lemma 2 [3] Let AUM,  , and A be a Herimite matrix,U be any unitary matrix, dd1,, n be the H principal diagonal elements of matrix A , 1,, n be the eigenvalues of , then dUAU  A.

2. Main Results Theorem 1 Let A and B be n n complex matrices, and W be mm unitary matrix, then m H H (2.1) RetrWAA   Rei W i AA , i1 m HH (2.2) Imtr WAA   Imii ( W ) ( AA ). i1 Proof. In general, we assume mn . The singular value decomposition of A is A  U,OV H , where U is a unitary matrix, V is a nn unitary matrix,

  diag1 A,, m A, 1 A    m A  0, and O is a mn  m null matrix.

Let V  V1,V2 , where V1 is a nm matrix, V2 is a n() n m matrix, obviously, V1 and V2 column H H unitary matrices, and VVI11 m , VVI22 nm . V H  Thus, A  U,O 1   UV H . Let H , E UV H ,then is a Semi-positive  H  1 GUU 1 G Herimite V2  definite matrix, and EEH  I , for any unitary matrix W ,so we have WAAH  WGEE H G WG 2 WGW HWG. Let W  Q H Q, where Q is a mm unitary matrix,  is a diagonal matrix , decompose  into the sum of real and imaginary parts,   diag1  W,,. n W   R   I

Where  R is the real partial diagonal matrix of  ,  I is the imaginary partial diagonal matrix .then

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Yuanyuan TU & Runqing SU Journal of Scientific and Engineering Research, 2019, 6(7):89-93

H H H WAA  WGW Q  R   I QG . H H H H  WGW Q  RQG WGW Q  I QG. Hence Retr WAAHHHH tr WGW  Q  QG  tr QW G QWH  QGQ    RR       mm HHHH    i  RQGQ QWG QW   i  R  QGQ QWG QH   ii11    mm HHHH i  RQGQ QWG QW    i  R  i QGQ  i  QWG QW   ii11 mm 2 H ReiW  i G Re  i W  i  AA  . ii11

So the inequality (2.1) holds, inequality (2.2) holds similarly. The proof is completed. Remark 1 if V  A  I, B  AA H , (1.1) yields the following inequality n HH Retr ( WAA )  Reii ( W )  AA  i1 n H   Reii (W ) AA  . i1 When H , i AA   1 n HH Retr WAA   Reii W  AA  i1

n H   ReiiW  AA  . i1 Under certain conditions, the inequality (2.1) is more accurate than (1.2). Thus, the Theorem 1 extended the conclusion of [2].

Theorem 2 Let AB, be Herimite matrices，U be a unitary matrix, i ()A , i ()B are the eigenvalue of A and B respectively, then n maxtr ( UAUH B )  ( A ) ( B ), (2.3) UUIH   ii i1 n mintr ( UAUH B )  ( A ) ( B ), (2.4) UUIH   i n i 1 i1 where 1 A   n A,1 B   n B. Proof. We first show (2.3). For is a matrix, so exists a unitary matrix V such that H H A  V1V  1  V AV , where 1  diag1 A,,n A.Similarly, exists a unitary matrix W such that H H W V BV W  2 , H where 2  diag1 B,,n B. Let T  VWV , notice that T is a unitary matrix, thus, we have H H H H H trTAT B trVWV V1V VW V B

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Yuanyuan TU & Runqing SU Journal of Scientific and Engineering Research, 2019, 6(7):89-93

n H H  tr1W V BVW   tr1 2   i Ai B. (2.5) i1 For any unitary matrix U , we show the following inequality holds n H tr( UAU B ) ii ( A ) ( B ). i1  ~ H ~ Denote B  (UV) B(UV)(bij ) , obviously , B is a Herimite matrix, and let bb1  n , where bi is the principal diagonal element of , hence n H ~ H H H (2.6) trUAU B trV1V U BU   tr1 UV BUV tr1B i Abii . i1 ~ Notice that B B.By Lemma 1 and Lemma 2, we have nn H (2.7) tr( UAU B )i ( A ) b i  i ( A )  i ( B ), ii11 By (2.5), (2.7), the equality (2.3) holds. Then we show (2.4). By Lemma 1, Lemma 2 and (2.6), and notice that ~ ~ dB B B.. So we have n n n trUAUB(H ) () Ab   () Ab   ()(). A  B i ii  i n i1  i ni1 i1 i  1 i  1 H Finally, let V be a unitary matrix and satisfy A  V1V , W be a unitary matrix which satisfy  H H W V BV W  2 . H where 1  diag1 A,,n A, 2  diagn B,,1 B. Let S  VWV , then S be a unitary matrix .Thus H H H H trSAS B trVWV V1V VW VB n H H  tr1W V BVW   tr12   i Ani1 B. i1 Hence, the equation (2.4) holds. Remark 2 if AB, are Semi-positive definite matrices, then

i AABB   i ,  i   i   . The equation (2.3) yields the following equation n H max H tr UAU B   A  B . (2.8) U U I    i   i   i1 However, Mirsky L gave the following equation in [4] n supRetrUAVB  suptrUAVB   i A i B, (2.9) U ,V U ,V i1 Obviously, (2.8) is a special case of (2.9), but (2.9) does not contain the Theorem 2. So, the Theorem 2 extended the conclusion of [4].

Acknowledgments Author: Tu yuanyuan, female, 1988, lecturer, master's degree, research direction: matrix, algebra. National natural science foundation (11601234), Jiangsu natural science foundation (BK20160571).

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Yuanyuan TU & Runqing SU Journal of Scientific and Engineering Research, 2019, 6(7):89-93

References [1]. J. Von Neumann (1937). Some matrix-inequalities and metrization of matric-space. Tomsk Univ. Rev. 1, 286-300. [2]. N. Komaroff (2008). Enhancements to the von Neumann trace inequality. Linear Algebra and its Applications, (428): 738-741. [3]. Marshall A W, Olkin I (1979). Theory of Majorization and its Applications. Academic, New York, 16: 4-93. [4]. Mirsky L (1975). A trace inequality of John von Neumann. Monatshefte für Mathematik, 79(4): 303- 306. [5]. Mirsk L (1959). On the trace of matrix products. Mathematische Nachrichten, 20(3-6):171-174. [6]. Chrétien S, Wei T (2015). Von Neumann's trace inequality for tensors. Linear Algebra and its Applications, 482: 149-157.

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