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Journal of Inequalities and Applications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Inequalities for eigenvalues of matrices Journal of Inequalities and Applications 2013, 2013:6 doi:10.1186/1029-242X-2013-6 Xiaozeng Xu (cquxuxz@163.com) Chuanjiang He (cquxuxz@163.com) ISSN 1029-242X Article type Research Submission date 5 July 2012 Acceptance date 14 December 2012 Publication date 4 January 2013 Article URL http://www.journalofinequalitiesandapplications.com/content/2013/1/ This peer-reviewed article can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2013 Xu and He This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. JIA_182_edited [12/27 09:37] 1/8 Inequalities for eigenvalues of matrices a, b 1 a Xiaozeng Xu , Chuanjiang He a. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P. R. China b. School of Mathematics and Statistics,Chongqing University of Technology, Chongqing, 400054, P. R. China Abstract The purpose of the paper is to present some inequalities for eigenvalues of positive semideﬁnite matrices. Keywords: singular values; eigenvalues; unitarily invariant norm Subject Classiﬁcation: MSC (2010) 15A18, 15A60 1. Introduction Throughout this paper, Mn denotes the space of n × n complex matrices and Hn denotes the set of all Hermitian matrices in Mn . Let A, B ∈ Hn ; the order relation A ≥ B means, as usual, that A − B is positive semideﬁnite. We always denote the singular values of A by s1 (A) ≥ · · · ≥ sn (A). If A has real eigenvalues, we label them as λ1 (A) ≥ · · · ≥ λn (A). Let · denote any unitarily invariant norm on Mn . We denote by |A| the absolute value 1 operator of A, that is, |A| = (A∗ A) 2 , where A∗ is the adjoint operator of A. For positive real number a, b, the arithmetic-geometric mean inequality 1 Corresponding author. E-mail address: cquxuxz@163.com (X. Xu). Preprint submitted to Journal of Inequalities and Applications December 27, 2012 JIA_182_edited [12/27 09:37] 2/8 says that √ a+b ab ≤ . 2 It is equivalent to 2m a+b (ab)m ≤ , m = 1, 2, · · · . (1.1) 2 Let A, B ∈ Mn be positive semideﬁnite. Bhatia and Kittaneh [1] proved that for all m = 1, 2, · · ·, 2m A+B λj ((AB) ) ≤ λj m . (1.2) 2 This is a matrix version of (1.1). For more information on matrix versions of the arithmetic-geometric mean inequality, the reader is referred to [1-11] and the references therein. It is easy to see that the arithmetic-geometric mean inequality is also equivalent to 2/3 a+b a3/4 b3/4 ≤ . (1.3) 2 As pointed out in [10, p.198], although the arithmetic-geometric mean in- equalities can be written in diﬀerent ways and each of them may be obtained the other, the matrix versions suggested by them are diﬀerent. In this note, we obtain a reﬁnement of (1.2) and a log-majorization in- equality for eigenvalues. As an application of our result, we give a matrix version of (1.3). 2. Main results We begin this section with the following lemma, which is a question posed by Bhatia and Kittaneh [1](see also [8, 10]) and settled in the aﬃrmative by 2 JIA_182_edited [12/27 09:37] 3/8 Drury in [2]. Lemma 2.1. Let A, B ∈ Mn be positive semideﬁnite. Then 2 A+B sj (AB) ≤ sj . 2 As a consequence of Lemma 2.1, we have 1 |AB|1/2 ≤ A+B . (2.1) 2 It is a matrix version of the arithmetic-geometric mean inequality. By prop- erties of the matrix square function, we know that this last inequality is stronger than the assertion 2 A+B AB ≤ , 2 which is due to Bhatia and Kittaneh [1] and is also a matrix version of (1.1). Theorem 2.1. Let A, B ∈ Mn be positive semideﬁnite. Then for all m = 1, 2, · · ·, 2m A + B + A1/2 B 1/2 + B 1/2 A1/2 λj ((AB) ) ≤ λj m . (2.2) 4 Proof. By Lemma 2.1, we have m m λj A2 B 2 = λj A2 B 2 m = λj AB 2 A = (sj (AB))2m 4m (2.3) A+B ≤ sj 2 4m A+B = λj . 2 3 JIA_182_edited [12/27 09:37] 4/8 Replacing A, B by A1/2 , B 1/2 in (2.3), we have 2m A + B + A1/2 B 1/2 + B 1/2 A1/2 λj ((AB) ) ≤ λj m . 4 This completes the proof. Remark 2.1. Let A, B ∈ Mn be positive semideﬁnite. Note that 2 A1/2 − B 1/2 A + B A + B + A1/2 B 1/2 + B 1/2 A1/2 0≤ = − . 2 2 4 Therefore, the inequality (2.2) is a reﬁnement of the inequality (1.2). Remark 2.2. For m = 1, by (1.2), we have 2 A+B λj (AB) ≤ λj . (2.4) 2 For m = 1, by (2.2), we have 4 A+B λj A B 2 2 ≤ λj . (2.5) 2 In view of the inequalities (2.4) and (2.5), one may ask whether it is true that 2m A+B λj (A B ) ≤ λj m m (2.6) 2 for all m = 1, 2, · · ·. The answer is no. For m = 3, the inequality (2.6) is refuted by the following example: ⎡ ⎤ ⎡ ⎤ 5 −1 6 −4 A=⎣ ⎦, B = ⎣ ⎦. −1 9 −4 5 Theorem 2.2. Let A, B ∈ Mn be positive semideﬁnite. Then k k 3 Av B 1−v + A1−v B v A+B λj A B ≤ λj . j=1 2 j=1 2 4 JIA_182_edited [12/27 09:37] 5/8 Proof. By Weyl’s inequality, Horn’s inequality and Lemma 2.1, we have k k |λj (AXB)| = |λj (XAB)| j=1 j=1 k ≤ sj (XAB) j=1 k (2.7) ≤ sj (X) sj (AB) j=1 k k 2 A+B ≤ sj (X) sj . j=1 j=1 2 Putting Av B 1−v + A1−v B v X= , 0 ≤ v ≤ 1, 2 in (2.7) gives k k k 2 Av B 1−v + A1−v B v Av B 1−v + A1−v B v A+B λj A B ≤ sj sj . j=1 2 j=1 2 j=1 2 (2.8) In response to a conjecture by Zhan [11], Audenaert [3] proved that if 0 ≤ v ≤ 1, then Av B 1−v + A1−v B v A+B sj ≤ sj . (2.9) 2 2 1 The special case where v = 2 was obtained earlier in [6, 12] and the special 1 case where v = 4 was obtained earlier in [15]. It follows from (2.8) and (2.9) that k k 3 Av B 1−v + A1−v B v A+B λj A B ≤ λj . j=1 2 j=1 2 This completes the proof. Remark 2.3. As an application of Theorem 2.2, we now present a matrix 5 JIA_182_edited [12/27 09:37] 6/8 1 version of (1.3). Taking v = 2 in this last inequality, we have k k 3 A+B λj A 3/2 B 3/2 ≤ sj j=1 j=1 2 and so k k 3/2 A+B sj A3/4 B 3/4 ≤ sj , j=1 j=1 2 which is equivalent to k k 3/4 2/3 A+B sj A 3/4 B ≤ sj . j=1 j=1 2 Since weak log-majorization is stronger than weak majorization, we have k k 3/4 2/3 A+B sj A 3/4 B ≤ sj . j=1 j=1 2 By Fan’s dominance theorem [4, p.93], we get 2/3 1 A3/4 B 3/4 ≤ A+B . (2.10) 2 This is a matrix version of (1.3). Next, we give another proof of the inequality (2.10). Araki [13] (also see [14]) obtained the following log-majorization inequality: k k q/p sj Ap/2 B p Ap/2 ≤ sj Aq/2 B q Aq/2 , 0 < p ≤ q. (2.11) j=1 j=1 Putting 3 p= , q=2 2 in (2.11) gives k k 3/4 1/3 1/4 sj A 3/4 B 3/2 A ≤ sj AB 2 A , j=1 j=1 6 JIA_182_edited [12/27 09:37] 7/8 and so k k 3/4 2/3 sj A 3/4 B ≤ sj |AB|1/2 . j=1 j=1 By Fan’s dominance theorem [4, p.93], we get 2/3 A3/4 B 3/4 ≤ |AB|1/2 . (2.12) It follows from (2.1) and (2.12) that 2/3 1 A3/4 B 3/4 ≤ A+B . 2 References [1] R. Bhatia, F. Kittaneh. Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl. 308 (2000) 203-211. [2] S. W. Drury. On a question of Bhatia and Kittaneh. Linear Algebra Appl. 437 (2012) 1955-1960. [3] K. M. R. Audenaert. A singular value inequality for Heinz means. Linear Algebra Appl. 422 (2007) 279-283. [4] R. Bhatia. Matrix Analysis. Springer-Verlag, New York, 1997. [5] R. Bhatia, C. Davis. More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 14 (1993) 132-136. [6] R. Bhatia, F. Kittaneh. On the singular values of a product of operators. SIAM J. Matrix Anal. Appl. 11 (1990) 272-277. 7 JIA_182_edited [12/27 09:37] 8/8 [7] R. Bhatia. Interpolating the arithmetic-geometric mean inequality and its operator version. Linear Algebra Appl. 413 (2006) 355-363. [8] R. Bhatia, F. Kittaneh. The matrix arithmetic–geometric mean inequal- ity revisited. Linear Algebra Appl. 428 (2008) 2177-2191. [9] H. Kosaki. Arithmetic-geometric mean and related inequalities for op- erators. J. Funct. Anal. 156 (1998) 429-451. [10] R. Bhatia. Positive Deﬁnite Matrices. Princeton University Press, Princeton, 2007. [11] X. Zhan. Matrix Inequalities. Lecture Notes in Mathematics, vol.1790, Springer-Verlag, Berlin, 2002. [12] T. Ando. Matrix Young inequalities. Oper. Theory Adv. Appl. 75 (1995) 33-38. [13] H. Araki. On an inequality of Lieb and Thirring. Lett. Math. Phys. 19 (1990) 167-170. [14] F. Hiai. Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization. Interdisciplinary Information Sciences. 16 (2010) 139- 248. [15] Y. Tao. More results on singular value inequalities of matrices. Linear Algebra Appl. 416 (2006) 724-729. 8

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