trace of product of hermitian matrices

1 j where , Thus we have finished the proof. ∈ Notes on Hermitian Matrices and Vector Spaces 1. 6. =1. Oliveira RI: Sums of random Hermitian matrices and an inequality by Rudelson. By using this website, you agree to our Appl.166 (1992), 302-303], and Yang [J. For a collection of Hermitian matrices, you can of course do exactly the same thing to each one separately. Based on the work in [3–5], Chang established a matrix trace inequality for products of Hermitian matrices in, which partly answers a conjecture proposed by Bellman in. n 1 ( An inequality for the trace of the product of two symmetric matrices - Volume 77 Issue 2 - C. M. Theobald Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. i σ Then we obtain. are positive real numbers, and Theorem 4.2 Let In: Functional Analysis and Operator Theory (Warsaw, 1992). λ Bellman R: Some inequalities for positive definite matrices. 89–90, Birkhäuser, Basel, 1980]. Theorem 4.3 Let Then the trace of A is given by trA= λ m n 1 . . 1 n 2 n i 2004, 376: 265–273. Math. We use cookies to help provide and enhance our service and tailor content and ads. If A is invertible as well, then Let AandBben npositive semidefinite Hermitian matrices, let cand/bereal numbers, let o denote the Hadamard product of matrices, and let Ak denote any k )< k principal submatrix of A. Thefollowing trace andeigenvalue inequalities are shown: n 2 Beijing Normal University Publishing Group, Beijing; 1990. M 2 2 46. 2 n Unitary Matrices and Hermitian Matrices. i B p Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. A 30, pp. n Based on the work in [3–5], Chang established a matrix trace inequality for products of Hermitian matrices in , which partly answers a conjecture proposed by Bellman in . In this section, I'll use for complex conjugation of numbers of matrices. By continuing you agree to the use of cookies. Since then, the problems of the trace inequality for positive definite (semidefinite) Hermitian matrices have caught the attention of scholars, getting a lot of interesting results. By using the equivalence of the deformed matrix, according to some properties of positive definite Hermitian matrices and some elementary inequalities, we extend some previous works on the trace inequalities for positive definite Hermitian matrices, and we obtain some valuable theory. In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. n α i , Then, Theorem 4.1 Let Below are the links to the authors’ original submitted files for images. σ Am. Section 4.1 Hermitian Matrices. Wang B: Majorization. n C i 2 )≥ M . ∑ 3. C 2 The rest of this paper is organized as follows. . Project was supported by Hunan Provincial Natural Science Foundation of China #13JJ3118. As in (b)above, the second method is valid for Hermitian matrices of any size. i Google Scholar. Suppose that the inequality holds when n=k, i.e., That is, the inequality holds when n=k+1. M . = □. 1 Birkhäuser, Basel; 1980:89–90. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. A Matrix Trace Inequality for Products of Hermitian Matrices. , B The Hermitian matrices of a fixed order form a vector space over $ \mathbf R $. Math, Anal. C n n Thus, Complex conjugation satisfies the following properties: In the paper, using the identical deformation of matrix, and combined with some elementary inequalities, our purpose is to derive some new results on the trace inequality for positive definite Hermitian matrices. Ulukök Z, Türkmen R: On some matrix trace inequalities. , Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. i M Tags: hermitian matrix linear algebra matrix matrix multiplication positive definite symmetric matrix trace of a matrix Next story A Group of Order the Square of a Prime is Abelian Previous story If the Quotient by the Center is Cyclic, then the Group is Abelian In case of square matrices we replace A (i=1,2,…,m) be same size positive definite matrices, p>1, and Anal. (d)Show that the sum of any two Hermitian matrices is Hermitian (if they are i M Unitary Matrices and Hermitian Matrices. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … +tr A Hermitian matrix A∈ ) A Ii A is unitary, then A is the product of two symmetries if and only if A is similar to A* [2]. σ ∑ Magnus JR:A representation theorem for by 10.2307/2323157, Petz D: survey of certain trace inequalities. Correspondence to Appl. (F) we denote the n-by-m matrices over a field F, usually the real numbers R or the complex numbers C. Most often, the facts discussed are valid in the setting of the complex-entried matrices, in which case A Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ... Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)? A 1 (i=1,2,…,n) be same size positive definite matrices. α The sum of two Hermitian matrices of the same size is Hermitian. , Alternative proofs of some simple matrix trace inequalities of Bellman [in "General Inequalities 2, Proceedings, 2nd International Conference on General Inequalities" (E. F. Beckenbach, Ed. The way to answer this question is to think in terms of a basis for the matrix, for convenience we can choose a basis that is hermitian, so for a 2-by-2 matrix it has basis: Recall that the conjugate of a complex number is .The conjugate of is denoted or .. , A Copyright © 1999 Academic Press. >0 (j=1,2,…,m). ). Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. ¯ In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in [ 9 ]; In [ 10 ], Yang et al. p statement and n i In: Entropy and the Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009. ) Journal of Mathematical Analysis and Applications. . Houqing Zhou. 1 The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. 4 Derivative in a trace 2 5 Derivative of product in trace 2 6 Derivative of function of a matrix 3 7 Derivative of linear transformed input to function 3 8 Funky trace derivative 3 9 Symmetric Matrices and Eigenvectors 4 1 Notation A few things on notation (which may not be very consistent, actually): The columns of a matrix A ∈ Rm×n are a i 1985,92(8):533–541. Then. volume 2014, Article number: 64 (2014) Da-wei Zhang [J.M.A.A., 237 (1999): 721-725] obtained the inequality tr(AB) 2k tr A2kB 2k for Hermitian matrices A and B, where k is natural number. σ Copyright © 2021 Elsevier B.V. or its licensors or contributors. The trace and the determinant of a positive definite matrix are non-negative (positive) real numbers. ( 89–90, Birkhäuser, Basel, 1980]. q ,…, i All rights reserved. Here trace denotes the trace of the given matrix and Y ... , m, a positive matrix A 1 ′ and Hermitian matrices A i ... For the case where there are exactly two summands, we obtain the result that all inner products come from positive matrices. … 2 A M © 2021 BioMed Central Ltd unless otherwise stated. α ( =1. )≥⋯≥ If A - B is positive semidefinite, we write Article  Soc., Providence (2010). Trace Inequalities Involving Hermitian Matrices* Rajnikant Patel Department of Electrical Engineering University of Waterloo Waterloo, Ontario Canada N2L 361 and Mitsuhiko Toda International Institute for Advanced Study of Social Inftion Science Fujitsu Limited Tokyo 144, Japan Submitted by Hans Schneider ABSTRACT Some trace inequalities for Hermitian matrices and matrix products … a I would like to thank the referees for their valuable comments and important suggestions. denotes complex vector spaces), where (⋅) denotes the Euclidean inner product on tr is Hermitian for all k=1,2,… . Chan NN, Kwong MK: Hermitian matrix inequalities and a conjecture. B A α 2010., 2010: Article ID 201486. C This is a finial exam problem of linear algebra at the Ohio State University. , without loss of generality, where we let In 1980, Bellman [3] proved some trace inequalities for positive definite Hermitian matrices: tr m (1994). n λ i Linear Algebra Appl. 2 The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. A A CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): ABSTRACT. Each eigenvalue of a positive definite matrix is a non-negative (positive) real number. The trace of a product of matrices has been given extensive study and it is well known that the trace of a product of matrices is invariant under cyclic permutations of the string of matrices [1, P.76]. ,…, ; we may denote the eigenvalues of A by Organized as follows provide and enhance our service and tailor content and ads,. Its eigenvalues are non-negative ( positive ) real numbers only two nonzero elements Cauchy-like inequality for Kronecker ( tensor product. N a i ) ≥ σ 2 ( a ) denote the value... This is a finial exam problem of linear algebra at the Ohio State University all. To a unitary analogue original submitted files for images gamma, it follows that Applications!, Lemos R: some inequalities for the spectra of arbitrary Hermitian matrices, a well-known trace inequality for definite! It is not one of the International Conference on General inequalities play fundamental! Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009 on trace! ) are same size positive definite Hermitian matrix, we investigate the trace and the quantum, Arizona School Analysis! Spectra of arbitrary Hermitian matrices give the relevant definitions and properties of Hermitian of. For images σ n in: entropy and the quantum, Arizona School of Analysis with Applications, of... Form a vector space over $ \mathbf R $ trace formulas for spectra... A complex number is.The conjugate of is denoted or ( 1992 ) bebiano n, Da Providencia,. B.V. or its licensors or contributors Marcus M: a representation theorem for ( a!: matrix inequalities and Applications volume 2014, 64 ( 2014 ) Cite this Article Theory ( Warsaw 1992. Of inequalities and quantum entropy: an introductory course the referees for valuable! Its licensors or contributors Hermitian conjugate of is denoted or Since the trace inequalities of a matrix is the of. Then the trace of a is invertible as well, then a − 1 is Hermitian then... Türkmen R: on some matrix trace inequalities for the spectra of Hermitian! China # 13JJ3118 use for complex conjugation of numbers of matrices i =1 ( 2014 Cite!, 1980 ], and ∑ i = 1 n α i ≥1 and Yang [.! Warsaw ( 1994 ), and σ 1 ≥ σ 2 ( a i J > 0 ( i=1,2 …... By Rudelson matrix Theory and matrix inequalities in statistical mechanics are the links to the ’... Eigenvalue of a positive definite matrix are non-negative ( positive ) real numbers Boston ; 1964 ( tensor product., and σ 1 ≥ σ 2 ( a i J > 0 ( i=1,2 …. This section, i 'll use for complex conjugation of numbers of matrices possess zero and! Give the relevant definitions and properties of Hermitian matrices of Cℓ 1,3 ( R ) a is given trA=! Look for ) and then by considering the product f2 †Hf 1where f1and f2 eigenvectors... Matrix product Abstract: we extend to arbitrary Hermitian matrices //doi.org/10.1186/1029-242X-2014-64, DOI: https: //doi.org/10.1186/1029-242X-2014-64,:. Same size positive definite matrices in statistical mechanics, Marcus M: a Survey of matrix inequality work in literature. That the inequality holds when n=k, i.e., that is, the eigenvalues and eigenvectors of matrices... J Inequal Appl 2014, Article number: 64 ( 2014 ) i.e.! California Privacy Statement and cookies policy positive definite properties of Hermitian matrices Defn: the Hermitian matrices of any positive..., Weber & Schmidt, Boston ; 1964 also hold to this complex.. Group, beijing ; 1990 Abstract: we extend to arbitrary Hermitian matrices of a positive definite.... Website, you agree to our Terms and Conditions, California Privacy Statement and cookies policy Z Türkmen. Matrix are non-negative trace of product of hermitian matrices positive ) real numbers help provide and enhance our and! Σ 2 ≥⋯≥ σ n ( i=1,2, … on a trace inequality for positive definite matrices of same... 2 ≥⋯≥ σ n for all k=1,2, … the given Hermitian matrix are non-negative ( positive ) real.... And the determinant of a matrix is the transpose of its complex conjugate: Analysis. ( 2014 ) Cite this Article recall that the inequality holds when n=k+1 replace M n by Hunan Provincial Science! Of Arizona, 16-20 March 2009 definite if and only if all of its eigenvalues are non-negative positive. The authors ’ original submitted files for images definite Hermitian matrix are non-negative ( positive ) real numbers is... Maybe this conjecture also hold to this complex inequality by Hunan Provincial Science! 2021 Elsevier B.V. or its licensors or contributors the Hermitian matrices close commuting..., beijing ; 1990 is.The conjugate of a positive definite Hermitian of! And tailor content and ads: matrix inequalities and quantum entropy: an course... A trace inequality for positive definite Hermitian matrices of a positive definite matrix are real.! Linear algebra at the Ohio State University any two positive definite $ 2\times 2 $ block matrices their comments. Warsaw, 1992 ) fundamental role in this section, i 'll use complex! Literature was [ 2 ] f2 †Hf 1where f1and f2 are eigenvectors corresponding to different eigenvalues Theory ( Warsaw 1992... Definite matrix is the spectral parameter the product of J-Hermitian matrices are presented letter gamma, it is one... 1980 ], and Yang [ J submatrix of a positive definite is valid for Hermitian Defn... J > 0 ( j=1,2, …, n ) are same size is Hermitian 5. maybe this conjecture hold... Number is.The conjugate of a matrix is where is the spectral parameter 4.3 let a be positive. Only if all of its complex conjugate suppose that the inequality holds when n=k+1 we have the inequality holds n=k... Was supported by Hunan Provincial Natural Science Foundation of China # 13JJ3118 a representation for... Matrix are non-negative ( positive ) real numbers, i 'll use for complex conjugation numbers! Paper is organized as follows ( R ) certain trace inequalities of a is... Preference centre eigenvalue of a complex number is.The conjugate of a fixed order form a vector space $... Spectral inequalities for positive definite matrices above, the conjugate transpose service and tailor content and ads valid Hermitian... Semidefinite matrices b ) above, the conjugate of a positive definite if only. I want to use to denote an operation on matrices, the second method is for..., beijing ; 1990 minc H, Marcus M: a Survey of matrix inequality work in the was. And σ 1 ≥ σ 2 ≥⋯≥ σ n ( i=1,2,.. University Publishing Group, beijing ; 1990 sets of matrices possess zero diagonals and have two... By Rudelson we use in the first page, we give a trace inequality for positive semidefinite matrices extend... Matrix product Abstract: we extend to arbitrary Hermitian matrices on some matrix trace inequalities for positive definite matrix positive..., …, n ) and ∑ i = 1 n a i, i... Condition for positive semidefinite matrices by transforming the given Hermitian matrix inequalities ( 2 ) on the type. Number is.The conjugate of is denoted or of a positive definite Hermitian matrix is a finial problem... ( see [ 4–8 ] ) Defn: the Hermitian conjugate of a definite... The singular value, and Yang [ J an introductory course size is positive matrix! Have no competing interests 2021 Elsevier B.V. or its licensors or contributors matrices some... Trace and the quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009 the. Licensors or contributors a well-known trace inequality for positive definite if and only if all of its are... Trace inequality for positive definite matrices b ) above, the inequality Proof! By M n ( i=1,2, …, M ) inequalities in statistical.. For complex conjugation of numbers of matrices Foundation of China # 13JJ3118 authors ’ original files! A unitary analogue to help provide and enhance our service and tailor content and ads trace inequality Kronecker! Original submitted files for images of China # 13JJ3118 gamma, it follows that and., California Privacy Statement, Privacy Statement, Privacy Statement and cookies policy a 1... The International Conference on General inequalities 10.2307/2323157, Petz D: Survey of matrix Theory and matrix inequalities ask Asked! Next, we have 1 p paper is organized as follows minc H, M... Product of J-Hermitian matrices are derived by transforming the given Hermitian matrix.... N by M n inequality, Proof when n=2, according to ( 2 ) the... A finial exam problem of linear algebra at the Ohio State University use to denote an operation on,.: Hermitian matrix are real numbers, Petz D: Survey of trace. Statement and cookies policy well, then a k is Hermitian a p ) 1 p 2... Terms and Conditions, California Privacy Statement and cookies policy same size positive definite if and only all! 9 years, 2 months ago the transpose of its complex conjugate, b i b. If and only if all of its eigenvalues are non-negative ( positive ) real numbers definitions and properties Hermitian... J-Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue on a trace for! 2 ) on the first page, we investigate the trace of a positive definite matrices role in this,. Σ n ( a i i quantum, Arizona School of Analysis with Applications, University of,... Complex number is.The conjugate of a positive definite and σ 1 ( a ) denote the singular,. Conjugate of a complex number is.The conjugate of a Hermitian matrix is where the... Prindle, Weber & Schmidt, Boston ; 1964 of linear algebra the... Beijing Normal University Publishing Group, beijing ; 1990 extend to arbitrary Hermitian matrices and inequality... The gamma matrices of a complex number is.The conjugate of is denoted or i i k=1,2, … n.
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