Some inequalities for sum and product of positive semide nite matrices

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1 Linear Algebra and its Applications 293 (1999) 39±49 Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b, *,3 a Department of Mathematics, Beijing Normal University, Beijing , People's Republic of China b Department of Mathematical Sciences, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FL 33314, USA Received 17 November 1998 accepted 16 December 1998 Submitted by R.A. Brualdi Abstract The purpose of this paper is to present some inequalities on majorization, unitarily invariant norm, trace, and eigenvalue for sum and product of positive semide nite (Hermitian) matrices. Some open questions proposed by Marshall and Olkin are resolved. Ó 1999 Elsevier Science Inc. All rights reserved. AMS classi cation 15A09 15A42 Keywords Majorization Eigenvalue Singular value Trace Unitarily invariant norm Inequality Moore±Penrose inverse Positive semide nite matrix 1. Introduction Let A be an n n complex matrix. Denote the eigenvalues of A by k 1 A k 2 A...k n A and singular values of A by r 1 A r 2 A...r n A, and let * Corresponding author. zhang@polaris.nova.edu 1 The work was supported in part by an NSF grant of China. 2 bywang@sun.ihep.ac.cn 3 The work was supported in part by the Nova Faculty Development Funds /99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII S ( 9 9 )

2 40 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 k A ˆ k 1 A k 2 A... k n A r A ˆ r 1 A r 2 A... r n A We further assume that the eigenvalues, if they are all real, and the singular values are arranged in decreasing order. As usual, we write A P 0 if A is positive semide nite (nonnegative de nite), A > 0 if A P 0 and A is nonsingular, and A P B if A B P 0 for Hermitian matrices A and B. An identity matrix is denoted by I. Throughout the paper we assume that all the matrices are n n unless otherwise stated. We rst revisit a Fan±Ho man inequality [3] or [4, p. 266] If A P 0, then for all unitary matrices U r A I w r A U w r A I Here w stands for weak majorization, that is, x w y means that every partial sum of the real vector x is dominated by the corresponding partial sum of the vector y, where x and y are real vectors with components arranged in decreasing order. Besides, we write x 6 y if x is dominated by y entrywise. We demonstrate that a more general version r A B w r A BU w r A B where A B P 0 and U is unitary, does not hold in general. But, with the middle term removed, it is true that for all A B P 0 r A B w r A B This will follow from a stronger log-majorization inequality (Theorem 1). We then turn our attention to answering some questions raised by Marshall and Olkin, generalizing the results on Euclidean norm to unitarily invariant norm. After this, in Section 4, we show the trace inequality that for any positive semide nite matrices A B and contraction matrices U V tr A B 6 tr ja UBV j 6 tr A B where jx j ˆ X X 1=2 (Theorem 3). In Section 5, we examine the eigenvalues of matrix product. Recall that if A P 0 then k A P k A 0 where A is any principal submatrix of A. This does not generalize to the product AB, where A B P 0, though, as is well known, the eigenvalues of AB are nonnegative (AB is not Hermitian in general). We have (Theorem 4), however, for any A > 0 and B P 0, k A 1 B P k A 1 B 0 In addition, we show (Theorem 6) that if A P 0 B P C P 0, then k A B B P k A C C

3 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39± Majorization inequality We adopt the notation log x w log y to mean that every partial product of x is less than or equal to the corresponding partial product of y, where x and y are vectors with nonnegative components in decreasing order, and use k k ui for any unitarily invariant norm on the matrix space. Theorem 1. Let A and B be positive semidefinite matrices of the same size. Then log r A B w log r A B 1 As a consequence r A B w r A B Thus ka Bk ui 6 ka Bk ui 2 3 Proof. First notice that for any positive semide nite A and B of the same size j det A B j 6 det A B by simultaneous congruence of A and B to diagonal matrices. Rewrite this inequality as Y n r i A B 6 Yn r i A B Since A B is Hermitian, there exists a unitary matrix U such that U A B U ˆ Diag d 1 d 2... d n with jd i j ˆ r i A B i ˆ n For each positive integer k 1 6 k 6 n let U 1 be the submatrix consisting of the rst k rows of U, and U 2 be the rest rows of U. Then by the above argument and the eigenvalue interlacing theorem Y k r i A B ˆ Yk 6 Yk r i U 1 A B U 1 r i U 1 A B U 1 6 Yk r i A B That is, log r A B w log r A B. Inequality (2) then follows since the log-majorization implies weak majorization [4, p. 7]. The norm

4 42 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 inequality is immediate due to the fact that r A w r B () kak ui 6 kbk ui [4, p. 264]. We note that r A B w r A BU w r A B does not hold in general for A B P 0 and unitary U. Take, for example, A ˆ 5 2 B ˆ 1 2 U ˆ Then r 1 A B ˆ 4 > r 1 A BU ˆ Questions of Marshall and Olkin This section aims to resolve the problems proposed by Marshall and Olkin in their book [4, ch. 10, Section B, pp. 269±270]. Denote A k ˆ Diag k 1 A k 2 A... k n A and A r ˆ Diag r 1 A r 2 A... r n A. The questions (below labeled as in the book) asked by Marshall and Olkin are whether the following results also hold for unitarily invariant norms B.8. Let A and B be the complex matrices, and U and V be unitary matrices satisfying the singular value decomposition B A ˆ U B A r V. Then for all unitary matrices C ka B UV k E 6 ka BCk E 4 Note that matrices A and B need not be real, and that there is a misprint in the book (second print) P in the inequality of B.8 should be 6. The proof of this inequality is a straightforward computation by writing the Euclidean norm as the square root of trace. B.8.a. Let A and B be complex matrices and let A ˆ U 1 A r V 1 and B ˆ U 2 B r V 2, where U 1 U 2 and V 1 V 2 are unitary matrices. Then for any unitary matrices U and V ka U 1 U 2 BV 2 V 1k E 6 ka UBV k E 5 Note that there are also misprints for this inequality in the book V 1 should be U 1 and U 1 be V 1. (The misprints in the two places have not been taken into account in [1,2].) B.9. Let A and B be complex normal matrices. Then for some permutation matrix P

5 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 43 min ka k VB k V k E ˆ ka k PB k P 0 k E 6 We now discuss whether these inequalities hold for unitarily invariant norms. For B.8, the answer is negative. Let, for example, A ˆ B ˆ C ˆ Then UV ˆ I 2 and r A B UV ˆ 6 6 w r A BC ˆ 10 0 In fact for any xed unitary matrix U 0, the general inequality ka BU 0 k ui 6 ka BCk ui cannot hold for all unitary matrices C. Suppose otherwise. We must have, on one hand, r 1 A BU r 1 A BU 0 r 2 A BU On the other hand, let H ˆ A BU 0 A BU 0 ˆ A 2 U 0 B2 U 0 7 U 0 U 0 ˆ R S T Then r 2 1 A BU 0 ˆ k 1 H P k 1 R k n S k 1 T P ˆ 36 It follows that r 1 A BU 0 ˆ 6 Notice also that r 2 1 A BU 0 r 2 2 A BU 0 ˆ tr H ˆ tr R tr S tr T ˆ 100 tr T P 72 Thus r 2 2 A BU 0 P 36, that is, r 1 A BU r 2 A BU P 12, a contradiction. Similarly, by noting that r A B I ˆ 8 8, we can prove that in general there is no unitary matrix V 0 such that for all unitary matrices C, ka BV 0 k ui P ka BCk ui For B.8.a, the answer is a rmative. Recall (cf. [5] or [7, p. 3]) that for any complex matrices A and B jr A r B j w r A B 7 where jxj ˆ jx 1 j jx 2 j... jx n j It follows that for any complex matrices A B and unitary matrices U V r A r B r ˆ jr A r B j w r A UBV w r A r B ˆ r A r B r Thus we have the following result.

6 44 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 Theorem 2. For any complex matrices A B, unitary matrices U V, and unitarily invariant norm k k ui, ka r B r k ui 6 ka UBV k ui 6 ka r B r k ui 8 The inequality in B.8.a for unitarily invariant norms follows at once, since ka U 1 U 2 BV 2 V 1k ui ˆ ku 1 AV 1 U 2 BV 2 k ui ˆ ka r B r k ui The inequalities in (8) may be rewritten in two-sided form as follows. For any unitary matrices U V ka U 0 BV 0 k ui 6 ka UBV k ui 6 ka U 0 BV 0 k ui where U 0 ˆ U 1 U 2 V 0 ˆ V 2 V 1. Likewise, for (4), one has for all unitary matrices C ka BW 0 k E 6 ka BCk E 6 ka BW 0 k E where W 0 ˆ UV and U V are the unitarily matrices in the polar decomposition B A ˆ U B A r V. The answer to B.9 for unitary invariant norm is negative. The question is equivalent to whether the inequality, given normal matrices A and B, ka k PB k P 0 k ui 6 ka k VB k V k ui 9 holds for some permutation matrix P and all unitary matrices V. For a counterexample, let A ˆ i 0 B ˆ 3 0 p p 1= 2 1= V ˆ p 2 0 i 0 1 1= p 2 1= 2 Then VBV ˆ r A k PB k P 0 ˆ w r A k VB k V ˆ Inequality (9), however, holds for Hermitian matrices. This is seen as follows If A and B are Hermitian matrices, then (see, e.g., [7, pp. 1, 50]) k A k B k A B and jk A k B j w jk A B j Using this, we have r A k B k ˆ jk A k B j ˆ jk A k k VB k V j w jk A k VB k V j ˆ r A k VB k V which implies (9) with P ˆ I.

7 What is more, one may prove the following identities Let A and B be normal matrices. Then min B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 45 ka VBV k E ˆ ka k PB k P 0 k E for some permutation matrix P max ka VBV k E ˆ ka k PB k P 0 k E for some permutation matrix P min ka VBV k E ˆ ka k PB k P 0 k E for some permutation matrix P max ka VBV k E ˆ ka k PB k P 0 k E for some permutation matrix P Note that none of the above identities holds in general if k k E is replaced by k k ui. For Hermitian matrices A and B, we have, by writing B k "ˆ Diag k n B... k 1 B min ka VBV k ui ˆ ka k B k k ui max ka VBV k ui ˆ ka k B k " k ui min ka VBV k ui ˆ ka k B k " k ui max ka VBV k ui ˆ ka k B k k ui 4. Trace inequality For any complex matrix X, we denote jx j ˆ X X 1=2. Recall that X is a contraction matrix if r 1 X 6 1 [8, p. 145] or [9, p. 154]. Note that unitary matrices are contractions. Theorem 3. Let A and B be positive semidefinite matrices. Then for any contraction matrices U and V tr A B 6 tr ja UBV j 6 tr A B 10 Proof. We rst show that if A P 0 then for any contraction matrices U and V Re tr A UAV P 0

8 46 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 To see this, let P be a unitary matrix such that A ˆ PA r P. Let P VUP ˆ R ˆ r ij Then jr ij j 6 1 and Re tr A UAV ˆ Re tr A AVU ˆ Re tr A r A r P VUP ˆ Xn r i A 1 Re r ii P 0 Note that inequality () still holds when the negative sign is replaced by the positive sign. Now let Q be a unitary matrix such that A UBV ˆ ja UBV jq (the polar decomposition). We have tr A B ˆ Re tr A B ˆ Re tr A UBV Re tr B UBV 6 Re tr A UBV for the second term is nonnegative by 6 trja UBV j ˆ Re tr AQ UBVQ ˆ tr A B Re tr A AQ Re tr B UBVQ 6 tr A B The inequality in the theorem may be rewritten as X n k i A B 6 Xn r i A UBV 6 Xn k i A B 5. Eigenvalue inequalities Let A and B be corresponding principal submatrices of positive semide nite matrices A and B, respectively. As is well known, the eigenvalues of AB and A B are all nonnegative. The eigenvalue interlacing theorem ensures that k A P k A 0 and k B P k B 0, where 0 is a zero matrix of appropriate size. But the inequality k AB P k A B 0 does not hold in general Take, for a counterexample, A ˆ and B ˆ with A ˆ 2 and B ˆ 1. We have, however, the following theorem.

9 Theorem 4. Let A > 0 B P 0. Then k A 1 B P k A 1 B 0 12 Proof. Let A be of size k k 1 6 k 6 n, and write A ˆ A A 12 A 21 A 22 Then B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 47 CAC ˆ A 0 0 fa C ˆ I k 0 A 21 A 1 I n k where fa ˆ A 22 A 21 A 1 A 12 is the Schur complement of A in A. Therefore A 1 ˆ C A C 0 fa Upon computation, we have! A 1 k A 1 0 B ˆ k!cbc 1 0 fa 00 1 A 1=2 0 ˆ A B! 0 1=2 AA 1=2 0 fa 1=2 0 fa!! A 1=2 B A 1=2 ˆ k P k A 1=2 B A 1=2 0 ˆ k A 1 B 0 where 's denote entries irrelevant to our discussions. The following theorem is a generalization of a result due to Patel and Toda [6] from trace to eigenvalue. The idea of the proof, given below for completion, is similar to that in [6]. Theorem 5. Let A P 0 B P C P 0 and A C > 0. Then k A B 1 B P k A C 1 C 13 Proof. Noticing that A B 1 6 A C 1, we have k A 1=2 A B 1 A 1=2 6 k A 1=2 A C 1 A 1=2

10 48 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 and k A B 1 B ˆ k I A B 1 A ˆ k I A 1=2 A B 1 A 1=2 P k I A 1=2 A C 1 A 1=2 ˆ k I A C 1 A ˆ k A C 1 C As a corollary [6] tr A B 1 B P tr A C 1 C The above theorem generalizes to Moore±Penrose g-inverses as follows. Theorem 6. Let A P 0 B P C P 0. Then k A B B P k A C C Proof. Let rank A B ˆ r and rank A C ˆ s. Then there exists a unitary matrix U such that UAU ˆ A1 0 UBU ˆ B1 0 UCU ˆ C where A 1 B 1 C 1 are of size r r, A 1 B 1 > 0 B 1 P C 1. Similarly, for some r r unitary matrix V VA 1 V ˆ A2 0 VB V ˆ B2 VC 1 V ˆ C where A 2 B 2 C 2 are of size s s, A 2 C 2 > 0 B 2 P C 2. Note that M 0 ˆ M 0 for any matrix M. We have k A B B ˆ k A 1 B 1 1 B 1 0 n r P k A 2 B 2 1 B 2 0 n s by Theorem 4 P k A 2 C 2 1 C 2 0 n s by Theorem 5 ˆ k A 1 C 1 C 1 0 n r ˆ k A C C As a corollary for A P 0 B P C P 0 tr A B B P tr A C C

11 B.-Y. Wang et al. / Linear Algebra and its Applications 293 (1999) 39±49 49 We end the paper by noting that (12) does not generalize to the Moore± Penrose g-inverses. Take, for a counterexample, A ˆ 1 1 B ˆ with A ˆ 1 and B ˆ 1. Then k A B ˆ 05 0 j k A B 0 ˆ 1 0 References [1] J. Bondar, Comments on and complements to ``Inequalities Theory of Majorization and its Applications'' by A.W. Marshall and I. Olkin. Linear Algebra and Its Applications 199 (1994) 5±129. [2] J. Bondar, Supplementary errata list for ``Inequalities Theory of Majorization and its Applications'' by A.W. Marshall and I. Olkin. private communication, [3] K. Fan, A. Ho man, Some metric inequalities in the space of matrices, Proc. Am. Math. Soc. 6 (1955) 1±6. [4] A.W. Marshall, I. Olkin, Inequalities Theory of Majorization and Its Applications, Academic Press, New York, [5] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford (2) (1960) 50±59. [6] R. Patel, M. Toda, Trace inequalities involving matrices, Linear Algebra Appl. 23 (1979) 13±20. [7] B.-Y. Wang, Introduction to Majorization and Matrix Inequalities, Beijing Normal University Press, 1990 (in Chinese). [8] F. Zhang, Matrix Theory Basic Results and Techniques, Springer, New York, [9] R. Horn, C. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991.

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