Some inequalities for sum and product of positive semide nite matrices
|
|
- Brittney Johnson
- 5 years ago
- Views:
Transcription
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.
Matrix Inequalities by Means of Block Matrices 1
Mathematical Inequalities & Applications, Vol. 4, No. 4, 200, pp. 48-490. Matrix Inequalities by Means of Block Matrices Fuzhen Zhang 2 Department of Math, Science and Technology Nova Southeastern University,
More informationSpectral inequalities and equalities involving products of matrices
Spectral inequalities and equalities involving products of matrices Chi-Kwong Li 1 Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187 (ckli@math.wm.edu) Yiu-Tung Poon Department
More informationInequalitiesInvolvingHadamardProductsof HermitianMatrices y
AppliedMathematics E-Notes, 1(2001), 91-96 c Availablefreeatmirrorsites ofhttp://math2.math.nthu.edu.tw/»amen/ InequalitiesInvolvingHadamardProductsof HermitianMatrices y Zhong-pengYang z,xianzhangchong-guangcao
More informationarxiv: v3 [math.ra] 22 Aug 2014
arxiv:1407.0331v3 [math.ra] 22 Aug 2014 Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms Abstract Chi-Kwong Li a, Fuzhen Zhang b a Department of Mathematics, College of William
More informationOn the Schur Complement of Diagonally Dominant Matrices
On the Schur Complement of Diagonally Dominant Matrices T.-G. Lei, C.-W. Woo,J.-Z.Liu, and F. Zhang 1 Introduction In 1979, Carlson and Markham proved that the Schur complements of strictly diagonally
More informationAbstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization relations.
HIROSHIMA S THEOREM AND MATRIX NORM INEQUALITIES MINGHUA LIN AND HENRY WOLKOWICZ Abstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization
More informationMath 408 Advanced Linear Algebra
Math 408 Advanced Linear Algebra Chi-Kwong Li Chapter 4 Hermitian and symmetric matrices Basic properties Theorem Let A M n. The following are equivalent. Remark (a) A is Hermitian, i.e., A = A. (b) x
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationYimin Wei a,b,,1, Xiezhang Li c,2, Fanbin Bu d, Fuzhen Zhang e. Abstract
Linear Algebra and its Applications 49 (006) 765 77 wwwelseviercom/locate/laa Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices Application of perturbation theory
More informationIntrinsic products and factorizations of matrices
Available online at www.sciencedirect.com Linear Algebra and its Applications 428 (2008) 5 3 www.elsevier.com/locate/laa Intrinsic products and factorizations of matrices Miroslav Fiedler Academy of Sciences
More informationb jσ(j), Keywords: Decomposable numerical range, principal character AMS Subject Classification: 15A60
On the Hu-Hurley-Tam Conjecture Concerning The Generalized Numerical Range Che-Man Cheng Faculty of Science and Technology, University of Macau, Macau. E-mail: fstcmc@umac.mo and Chi-Kwong Li Department
More informationSingular Value and Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017) Article 8 2017 Singular Value Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices Aliaa Burqan Zarqa University,
More informationInterpolating the arithmetic geometric mean inequality and its operator version
Linear Algebra and its Applications 413 (006) 355 363 www.elsevier.com/locate/laa Interpolating the arithmetic geometric mean inequality and its operator version Rajendra Bhatia Indian Statistical Institute,
More informationSchur complements and matrix inequalities in the Löwner ordering
Linear Algebra and its Applications 321 (2000) 399 410 www.elsevier.com/locate/laa Schur complements and matrix inequalities in the Löwner ordering Fuzhen Zhang Department of Mathematics, Sciences and
More informationCHARACTERIZATIONS. is pd/psd. Possible for all pd/psd matrices! Generating a pd/psd matrix: Choose any B Mn, then
LECTURE 6: POSITIVE DEFINITE MATRICES Definition: A Hermitian matrix A Mn is positive definite (pd) if x Ax > 0 x C n,x 0 A is positive semidefinite (psd) if x Ax 0. Definition: A Mn is negative (semi)definite
More informationON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES
ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES HANYU LI, HU YANG College of Mathematics and Physics Chongqing University Chongqing, 400030, P.R. China EMail: lihy.hy@gmail.com,
More informationCompound matrices and some classical inequalities
Compound matrices and some classical inequalities Tin-Yau Tam Mathematics & Statistics Auburn University Dec. 3, 04 We discuss some elegant proofs of several classical inequalities of matrices by using
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 6 (2012), no. 1, 139 146 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/bjma/ AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH
More informationClarkson Inequalities With Several Operators
isid/ms/2003/23 August 14, 2003 http://www.isid.ac.in/ statmath/eprints Clarkson Inequalities With Several Operators Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute, Delhi Centre 7, SJSS Marg,
More informationLinear estimation in models based on a graph
Linear Algebra and its Applications 302±303 (1999) 223±230 www.elsevier.com/locate/laa Linear estimation in models based on a graph R.B. Bapat * Indian Statistical Institute, New Delhi 110 016, India Received
More informationFuzhen Zhang s Publication List
Fuzhen Zhang s Publication List Papers in peer-reviewed journals 1. Inequalities of generalized matrix functions via tensor products, with Vehbi E. Paksoy and Ramazan Turkmen, Electron. J. Linear Algebra,
More informationLinear Operators Preserving the Numerical Range (Radius) on Triangular Matrices
Linear Operators Preserving the Numerical Range (Radius) on Triangular Matrices Chi-Kwong Li Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA. E-mail:
More informationON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES
olume 10 2009, Issue 2, Article 41, 10 pp. ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES HANYU LI, HU YANG, AND HUA SHAO COLLEGE OF MATHEMATICS AND PHYSICS CHONGQING UNIERSITY
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko
More informationMoore Penrose inverses and commuting elements of C -algebras
Moore Penrose inverses and commuting elements of C -algebras Julio Benítez Abstract Let a be an element of a C -algebra A satisfying aa = a a, where a is the Moore Penrose inverse of a and let b A. We
More informationNotes on matrix arithmetic geometric mean inequalities
Linear Algebra and its Applications 308 (000) 03 11 www.elsevier.com/locate/laa Notes on matrix arithmetic geometric mean inequalities Rajendra Bhatia a,, Fuad Kittaneh b a Indian Statistical Institute,
More informationarxiv: v1 [math.ca] 7 Jan 2015
Inertia of Loewner Matrices Rajendra Bhatia 1, Shmuel Friedland 2, Tanvi Jain 3 arxiv:1501.01505v1 [math.ca 7 Jan 2015 1 Indian Statistical Institute, New Delhi 110016, India rbh@isid.ac.in 2 Department
More informationFuzhen Zhang s Publication and Presentation List (updated Dec. 2007)
Fuzhen Zhang s Publication and Presentation List (updated Dec. 2007) Papers in peer-reviewed journals 1. Revisiting Hua-Marcus-Bellman-Ando inequalities on contractive matrices, with C. Xu and Z. Xu, Lin.
More informationPROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS
PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS Abstract. We present elementary proofs of the Cauchy-Binet Theorem on determinants and of the fact that the eigenvalues of a matrix
More informationSINGULAR VALUE INEQUALITIES FOR COMPACT OPERATORS
SINGULAR VALUE INEQUALITIES FOR OMPAT OPERATORS WASIM AUDEH AND FUAD KITTANEH Abstract. A singular value inequality due to hatia and Kittaneh says that if A and are compact operators on a complex separable
More informationOrthogonal similarity of a real matrix and its transpose
Available online at www.sciencedirect.com Linear Algebra and its Applications 428 (2008) 382 392 www.elsevier.com/locate/laa Orthogonal similarity of a real matrix and its transpose J. Vermeer Delft University
More informationarxiv: v1 [math.na] 1 Sep 2018
On the perturbation of an L -orthogonal projection Xuefeng Xu arxiv:18090000v1 [mathna] 1 Sep 018 September 5 018 Abstract The L -orthogonal projection is an important mathematical tool in scientific computing
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationPositive definite preserving linear transformations on symmetric matrix spaces
Positive definite preserving linear transformations on symmetric matrix spaces arxiv:1008.1347v1 [math.ra] 7 Aug 2010 Huynh Dinh Tuan-Tran Thi Nha Trang-Doan The Hieu Hue Geometry Group College of Education,
More informationMultiplicative Perturbation Bounds of the Group Inverse and Oblique Projection
Filomat 30: 06, 37 375 DOI 0.98/FIL67M Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Multiplicative Perturbation Bounds of the Group
More information642:550, Summer 2004, Supplement 6 The Perron-Frobenius Theorem. Summer 2004
642:550, Summer 2004, Supplement 6 The Perron-Frobenius Theorem. Summer 2004 Introduction Square matrices whose entries are all nonnegative have special properties. This was mentioned briefly in Section
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationSome inequalities for unitarily invariant norms of matrices
Wang et al Journal of Inequalities and Applications 011, 011:10 http://wwwjournalofinequalitiesandapplicationscom/content/011/1/10 RESEARCH Open Access Some inequalities for unitarily invariant norms of
More informationBounds for Levinger s function of nonnegative almost skew-symmetric matrices
Linear Algebra and its Applications 416 006 759 77 www.elsevier.com/locate/laa Bounds for Levinger s function of nonnegative almost skew-symmetric matrices Panayiotis J. Psarrakos a, Michael J. Tsatsomeros
More informationON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH
ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH V. FABER, J. LIESEN, AND P. TICHÝ Abstract. Numerous algorithms in numerical linear algebra are based on the reduction of a given matrix
More informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More informationThe matrix arithmetic-geometric mean inequality revisited
isid/ms/007/11 November 1 007 http://wwwisidacin/ statmath/eprints The matrix arithmetic-geometric mean inequality revisited Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute Delhi Centre 7 SJSS
More informationWavelets and Linear Algebra
Wavelets and Linear Algebra () (05) 49-54 Wavelets and Linear Algebra http://wala.vru.ac.ir Vali-e-Asr University of Rafsanjan Schur multiplier norm of product of matrices M. Khosravia,, A. Sheikhhosseinia
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal
More informationOn the second Laplacian eigenvalues of trees of odd order
Linear Algebra and its Applications 419 2006) 475 485 www.elsevier.com/locate/laa On the second Laplacian eigenvalues of trees of odd order Jia-yu Shao, Li Zhang, Xi-ying Yuan Department of Applied Mathematics,
More informationCHI-KWONG LI AND YIU-TUNG POON. Dedicated to Professor John Conway on the occasion of his retirement.
INTERPOLATION BY COMPLETELY POSITIVE MAPS CHI-KWONG LI AND YIU-TUNG POON Dedicated to Professor John Conway on the occasion of his retirement. Abstract. Given commuting families of Hermitian matrices {A
More informationELA ON A SCHUR COMPLEMENT INEQUALITY FOR THE HADAMARD PRODUCT OF CERTAIN TOTALLY NONNEGATIVE MATRICES
ON A SCHUR COMPLEMENT INEQUALITY FOR THE HADAMARD PRODUCT OF CERTAIN TOTALLY NONNEGATIVE MATRICES ZHONGPENG YANG AND XIAOXIA FENG Abstract. Under the entrywise dominance partial ordering, T.L. Markham
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationCentral Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J
Central Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J Frank Curtis, John Drew, Chi-Kwong Li, and Daniel Pragel September 25, 2003 Abstract We study central groupoids, central
More informationThe minimum rank of matrices and the equivalence class graph
Linear Algebra and its Applications 427 (2007) 161 170 wwwelseviercom/locate/laa The minimum rank of matrices and the equivalence class graph Rosário Fernandes, Cecília Perdigão Departamento de Matemática,
More informationA note on an unusual type of polar decomposition
A note on an unusual type of polar decomposition H. Faßbender a, Kh. D. Ikramov b,,1 a Institut Computational Mathematics, TU Braunschweig, Pockelsstr. 14, D-38023 Braunschweig, Germany. b Faculty of Computational
More informationa Λ q 1. Introduction
International Journal of Pure and Applied Mathematics Volume 9 No 26, 959-97 ISSN: -88 (printed version); ISSN: -95 (on-line version) url: http://wwwijpameu doi: 272/ijpamv9i7 PAijpameu EXPLICI MOORE-PENROSE
More informationTwo applications of the theory of primary matrix functions
Linear Algebra and its Applications 361 (2003) 99 106 wwwelseviercom/locate/laa Two applications of the theory of primary matrix functions Roger A Horn, Gregory G Piepmeyer Department of Mathematics, University
More informationRANKS OF QUANTUM STATES WITH PRESCRIBED REDUCED STATES
RANKS OF QUANTUM STATES WITH PRESCRIBED REDUCED STATES CHI-KWONG LI, YIU-TUNG POON, AND XUEFENG WANG Abstract. Let M n be the set of n n complex matrices. in this note, all the possible ranks of a bipartite
More informationMATH36001 Generalized Inverses and the SVD 2015
MATH36001 Generalized Inverses and the SVD 201 1 Generalized Inverses of Matrices A matrix has an inverse only if it is square and nonsingular. However there are theoretical and practical applications
More informationSome bounds for the spectral radius of the Hadamard product of matrices
Some bounds for the spectral radius of the Hadamard product of matrices Guang-Hui Cheng, Xiao-Yu Cheng, Ting-Zhu Huang, Tin-Yau Tam. June 1, 2004 Abstract Some bounds for the spectral radius of the Hadamard
More informationLecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why?
KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices Why? Lecture 5 Ch. 5, Norms for vectors and matrices Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Problem: Measure size of
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationConvexity of the Joint Numerical Range
Convexity of the Joint Numerical Range Chi-Kwong Li and Yiu-Tung Poon October 26, 2004 Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement. Abstract Let A = (A 1,..., A m ) be an
More informationan Off-Diagonal Block of
J. oflnequal. & Appl., 1999, Vol. 3, pp. 137-142 Reprints available directly from the publisher Photocopying permitted by license only (C) 1999 OPA (Overseas Publishers Association) N.V. Published by license
More informationAbsolute value equations
Linear Algebra and its Applications 419 (2006) 359 367 www.elsevier.com/locate/laa Absolute value equations O.L. Mangasarian, R.R. Meyer Computer Sciences Department, University of Wisconsin, 1210 West
More informationMatrix Completion Problems for Pairs of Related Classes of Matrices
Matrix Completion Problems for Pairs of Related Classes of Matrices Leslie Hogben Department of Mathematics Iowa State University Ames, IA 50011 lhogben@iastate.edu Abstract For a class X of real matrices,
More informationOptimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications
Optimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications Yongge Tian China Economics and Management Academy, Central University of Finance and Economics,
More informationMaximizing the numerical radii of matrices by permuting their entries
Maximizing the numerical radii of matrices by permuting their entries Wai-Shun Cheung and Chi-Kwong Li Dedicated to Professor Pei Yuan Wu. Abstract Let A be an n n complex matrix such that every row and
More informationA Characterization of the Hurwitz Stability of Metzler Matrices
29 American Control Conference Hyatt Regency Riverfront, St Louis, MO, USA June -2, 29 WeC52 A Characterization of the Hurwitz Stability of Metzler Matrices Kumpati S Narendra and Robert Shorten 2 Abstract
More informationOn Euclidean distance matrices
On Euclidean distance matrices R. Balaji and R. B. Bapat Indian Statistical Institute, New Delhi, 110016 November 19, 2006 Abstract If A is a real symmetric matrix and P is an orthogonal projection onto
More informationGeneralized Schur complements of matrices and compound matrices
Electronic Journal of Linear Algebra Volume 2 Volume 2 (200 Article 3 200 Generalized Schur complements of matrices and compound matrices Jianzhou Liu Rong Huang Follow this and additional wors at: http://repository.uwyo.edu/ela
More informationGeneralized Interlacing Inequalities
Generalized Interlacing Inequalities Chi-Kwong Li, Yiu-Tung Poon, and Nung-Sing Sze In memory of Professor Ky Fan Abstract We discuss some applications of generalized interlacing inequalities of Ky Fan
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 7, Issue 1, Article 34, 2006 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS ZEYAD AL
More informationThe Laplacian spectrum of a mixed graph
Linear Algebra and its Applications 353 (2002) 11 20 www.elsevier.com/locate/laa The Laplacian spectrum of a mixed graph Xiao-Dong Zhang a,, Jiong-Sheng Li b a Department of Mathematics, Shanghai Jiao
More informationHUA S MATRIX EQUALITY AND SCHUR COMPLEMENTS
INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 3, Number 1, Pages 1 18 c 2007 Institute for Scientific Computing and Information HUA S MATRIX EQUALITY AND SCHUR COMPLEMENTS CHRIS PAGE,
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 36 (01 1960 1968 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa The sum of orthogonal
More informationFuzhen Zhang s Publication and Presentation List
Fuzhen Zhang s Publication and Presentation List Papers in peer-reviewed journals 1. Takagi factorization for quaternion matrices, with R. Horn, submitted to LAMA. 2. Book Review for Inequalities: Theory
More informationSymmetric Norm Inequalities And Positive Semi-Definite Block-Matrices
Symmetric Norm Inequalities And Positive Semi-Definite lock-matrices Antoine Mhanna To cite this version: Antoine Mhanna Symmetric Norm Inequalities And Positive Semi-Definite lock-matrices 15
More informationarxiv: v1 [math.fa] 1 Oct 2015
SOME RESULTS ON SINGULAR VALUE INEQUALITIES OF COMPACT OPERATORS IN HILBERT SPACE arxiv:1510.00114v1 math.fa 1 Oct 2015 A. TAGHAVI, V. DARVISH, H. M. NAZARI, S. S. DRAGOMIR Abstract. We prove several singular
More informationDiagonal and Monomial Solutions of the Matrix Equation AXB = C
Iranian Journal of Mathematical Sciences and Informatics Vol. 9, No. 1 (2014), pp 31-42 Diagonal and Monomial Solutions of the Matrix Equation AXB = C Massoud Aman Department of Mathematics, Faculty of
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 430 (2009) 532 543 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa Computing tight upper bounds
More informationUniversitext. Series Editors: Sheldon Axler San Francisco State University. Vincenzo Capasso Università degli Studi di Milano
Universitext Universitext Series Editors: Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus J. MacIntyre Queen
More informationarxiv: v1 [math.co] 10 Aug 2016
POLYTOPES OF STOCHASTIC TENSORS HAIXIA CHANG 1, VEHBI E. PAKSOY 2 AND FUZHEN ZHANG 2 arxiv:1608.03203v1 [math.co] 10 Aug 2016 Abstract. Considering n n n stochastic tensors (a ijk ) (i.e., nonnegative
More informationMinimum number of non-zero-entries in a 7 7 stable matrix
Linear Algebra and its Applications 572 (2019) 135 152 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Minimum number of non-zero-entries in a
More informationOn the adjacency matrix of a block graph
On the adjacency matrix of a block graph R. B. Bapat Stat-Math Unit Indian Statistical Institute, Delhi 7-SJSS Marg, New Delhi 110 016, India. email: rbb@isid.ac.in Souvik Roy Economics and Planning Unit
More informationJordan Canonical Form of A Partitioned Complex Matrix and Its Application to Real Quaternion Matrices
COMMUNICATIONS IN ALGEBRA, 29(6, 2363-2375(200 Jordan Canonical Form of A Partitioned Complex Matrix and Its Application to Real Quaternion Matrices Fuzhen Zhang Department of Math Science and Technology
More informationNorm Inequalities of Positive Semi-Definite Matrices
Norm Inequalities of Positive Semi-Definite Matrices Antoine Mhanna To cite this version: Antoine Mhanna Norm Inequalities of Positive Semi-Definite Matrices 15 HAL Id: hal-11844 https://halinriafr/hal-11844v1
More informationOn the Hermitian solutions of the
Journal of Applied Mathematics & Bioinformatics vol.1 no.2 2011 109-129 ISSN: 1792-7625 (print) 1792-8850 (online) International Scientific Press 2011 On the Hermitian solutions of the matrix equation
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics MATRIX AND OPERATOR INEQUALITIES FOZI M DANNAN Department of Mathematics Faculty of Science Qatar University Doha - Qatar EMail: fmdannan@queduqa
More informationUniversity, Fort Lauderdale, FL, USA. Published online: 30 Jun To link to this article:
This article was downloaded by: [172.4.212.45] On: 12 August 2014, At: 11:32 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer
More informationTrace Inequalities for a Block Hadamard Product
Filomat 32:1 2018), 285 292 https://doiorg/102298/fil1801285p Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Trace Inequalities for
More informationInequalities for the spectra of symmetric doubly stochastic matrices
Linear Algebra and its Applications 49 (2006) 643 647 wwwelseviercom/locate/laa Inequalities for the spectra of symmetric doubly stochastic matrices Rajesh Pereira a,, Mohammad Ali Vali b a Department
More informationInequalities For Singular Values And Traces Of Quaternion Hermitian Matrices
Inequalities For Singular Values And Traces Of Quaternion Hermitian Matrices K. Gunasekaran M. Rahamathunisha Ramanujan Research Centre, PG and Research Department of Mathematics, Government Arts College
More informationModified Gauss Seidel type methods and Jacobi type methods for Z-matrices
Linear Algebra and its Applications 7 (2) 227 24 www.elsevier.com/locate/laa Modified Gauss Seidel type methods and Jacobi type methods for Z-matrices Wen Li a,, Weiwei Sun b a Department of Mathematics,
More informationRITZ VALUE BOUNDS THAT EXPLOIT QUASI-SPARSITY
RITZ VALUE BOUNDS THAT EXPLOIT QUASI-SPARSITY ILSE C.F. IPSEN Abstract. Absolute and relative perturbation bounds for Ritz values of complex square matrices are presented. The bounds exploit quasi-sparsity
More informationThroughout these notes we assume V, W are finite dimensional inner product spaces over C.
Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal
More informationOperators with numerical range in a closed halfplane
Operators with numerical range in a closed halfplane Wai-Shun Cheung 1 Department of Mathematics, University of Hong Kong, Hong Kong, P. R. China. wshun@graduate.hku.hk Chi-Kwong Li 2 Department of Mathematics,
More informationThe Improved Arithmetic-Geometric Mean Inequalities for Matrix Norms
Applied Mathematical Sciences, Vol 7, 03, no 9, 439-446 HIKARI Ltd, wwwm-hikaricom The Improved Arithmetic-Geometric Mean Inequalities for Matrix Norms I Halil Gumus Adıyaman University, Faculty of Arts
More informationYongge Tian. China Economics and Management Academy, Central University of Finance and Economics, Beijing , China
On global optimizations of the rank and inertia of the matrix function A 1 B 1 XB 1 subject to a pair of matrix equations B 2 XB 2, B XB = A 2, A Yongge Tian China Economics and Management Academy, Central
More informationCompression, Matrix Range and Completely Positive Map
Compression, Matrix Range and Completely Positive Map Iowa State University Iowa-Nebraska Functional Analysis Seminar November 5, 2016 Definitions and notations H, K : Hilbert space. If dim H = n
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 431 (29) 188 195 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Lattices associated with
More informationMINIMAL NORMAL AND COMMUTING COMPLETIONS
INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 4, Number 1, Pages 5 59 c 8 Institute for Scientific Computing and Information MINIMAL NORMAL AND COMMUTING COMPLETIONS DAVID P KIMSEY AND
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationMatrix Theory, Math6304 Lecture Notes from October 25, 2012
Matrix Theory, Math6304 Lecture Notes from October 25, 2012 taken by John Haas Last Time (10/23/12) Example of Low Rank Perturbation Relationship Between Eigenvalues and Principal Submatrices: We started
More information