n M If λ, µ, ν are three spectra, we deﬁne the relation (1) λ⊞µ ∼ c ν if there exist Hermitian matrices H λ ∈ O λ, H µ ∈ O µ, H ν ∈ O ν such that H λ + H µ = H ν. 2 i Notes on Hermitian Matrices and Vector Spaces 1. 1 , A The trace function has the following properties. α PubMed Google Scholar. i 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. n α )≥ j n We use cookies to help provide and enhance our service and tailor content and ads. B California Privacy Statement, 2 n , 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. 2 A trace inequality for matrix product Abstract: We extend to arbitrary Hermitian matrices, a well-known trace inequality for positive semidefinite matrices. Soc., Providence (2010). A In this note, the following matrix trace inequality for products of Hermitian matrices A and B, tr AB 2 k ≤ tr A 2 k B 2 k, is established, where k is an integer. n A (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 Let Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. i ≥ The rest of this paper is organized as follows. Springer, Berlin; 1970. In linear algebra, the trace of a square matrix A, denoted (), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A.. i B Ask Question Asked 5 months ago. 2 ∑ statement and Introduction 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]. … 1 Commun. 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]. Basic definitions. ≥1. All rights reserved. C A complex matrix A is the product of two hermitian matrices if and only if it is similar to a real matrix [l ] ; this occurs if and only if A is similar to its adjoint [5]. Notes on Hermitian Matrices and Vector Spaces 1. , In the first type the unitary matrix is where is the spectral parameter. n ), pp. ; we may denote the eigenvalues of A by λ Let Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … $\endgroup$ – David Handelman Sep 25 '15 at 13:11 2 2 Mitrinovic DS, Vasic PP: Analytic Inequalities. ,…, Each eigenvalue of a positive definite matrix is a non-negative (positive) real number. α 2 Minc H, Marcus M: A Survey of Matrix Theory and Matrix Inequalities. is Hermitian for all k=1,2,… . 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 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. . Under the operation $ A \cdot B = ( AB + BA ) / 2 $ the Hermitian matrices (of order $ n $) form a Jordan algebra. https://creativecommons.org/licenses/by/2.0, https://doi.org/10.1186/1029-242X-2014-64. Beijing Normal University Publishing Group, Beijing; 1990. =1. The sum of any two positive definite matrices of the same size is positive definite. ≤tr( tr p i by Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. By continuing you agree to the use of cookies. In 1980, Bellman [3] proved some trace inequalities for positive definite Hermitian matrices: tr >0 (i=1,2,…,n), and A , . Both sets of matrices possess zero diagonals and have only two nonzero elements. (1994). A m 2 In this section, I'll use for complex conjugation of numbers of matrices. n Prindle, Weber & Schmidt, Boston; 1964. For example, i 1987, 95: 127–134. The eigenvalues and eigenvectors of Hermitian matrices have some special properties. As in (b)above, the second method is valid for Hermitian matrices of any size. ∑ Thus, Complex conjugation satisfies the following properties: A ( Then, Let . https://doi.org/10.1186/1029-242X-2014-64, DOI: https://doi.org/10.1186/1029-242X-2014-64. σ Math. n A α M A © 2021 BioMed Central Ltd unless otherwise stated. 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: 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. volume 2014, Article number: 64 (2014) trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. In Section 3, we will quote some lemmas; in Section 4, which is the main part of the paper, using the properties of Hermitian matrices, we investigate the trace inequalities for positive definite Hermitian matrices. , 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 the first type the unitary matrix is where is the spectral parameter. J Inequal Appl 2014, 64 (2014). . where Sci., Warsaw (1994), Carlen E: Trace inequalities and quantum entropy: an introductory course. If λ, µ, ν are three spectra, we deﬁne the relation (1) λ⊞µ ∼ c ν if there exist Hermitian matrices H λ ∈ O λ, H µ ∈ O µ, H ν ∈ O ν such that H λ + H µ = H ν. ≥⋯≥ 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 p The Hermitian matrices of a fixed order form a vector space over $ \mathbf R $. ( Probab. Ulukök Z, Türkmen R: On some matrix trace inequalities. Theorem 4.3 Let In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in [ 9 ]; In [ 10 ], Yang et al. A 1985,92(8):533–541. If A - B is positive semidefinite, we write 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 . λ Bellman R: Some inequalities for positive definite matrices. m Recall that the conjugate of a complex number is .The conjugate of is denoted or .. Let i Let H n denote the space of Hermitian n × n matrices, H n + denote the set consisting of positive semi-definite n × n Hermitian matrices and H n ++ denote the set of positive definite Hermitian matrices. a Contemporary Mathematics. M . So, for example, if M= 0 @ 1 i 0 2 1 i 1 + i 1 A; then its Hermitian conjugate Myis My= 1 0 1 + i i 2 1 i : In terms of matrix elements, [My] ij = ([M] ji): Note that for any matrix (Ay)y= A: (i=1,2,…,m). (i=1,2,…,m) be same size positive definite matrices, p>1, and . >0 (i=1,2,…,n) and i C Mon. n Linear Algebra Appl. A Hermitian matrix is positive definite if and only if all of its eigenvalues are non-negative (positive) real numbers. q Ii A is unitary, then A is the product of two symmetries if and only if A is similar to A* [2]. 287-298. σ ), pp. 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. The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. σ In: Entropy and the Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009. This is a finial exam problem of linear algebra at the Ohio State University. ∈ are positive real numbers, and 30, pp. Springer Nature. 1 Then. Oliveira RI: Sums of random Hermitian matrices and an inequality by Rudelson. Chan NN, Kwong MK: Hermitian matrix inequalities and a conjecture. When using this result for the product of two matrices when only one is Hermitian, we improve a recent trace inequality given by Fanf et al. . A ∑ I want to use to denote an operation on matrices, the conjugate transpose.. i Then we have the inequality, Proof When n=2, according to (2) on the first page, we have. A 2 Section 4.1 Hermitian Matrices. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. ... Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)? ) By i A Terms and Conditions, Am. 89–90, Birkhäuser, Basel, 1980]. Thus we have finished the proof. J. Inequal. So, for example, if M= 0 @ 1 i 0 2 1 i 1 + i 1 A; then its Hermitian conjugate Myis My= 1 0 1 + i i 2 1 i : In terms of matrix elements, [My] ij = ([M] ji): Note that for any matrix (Ay)y= A: In this note, the following matrix trace inequality for products of Hermitian matrices A and B,trAB2k≤trA2kB2k,is established, where k is an integer. Project was supported by Hunan Provincial Natural Science Foundation of China #13JJ3118. For example, Theorem 2.4 applies to C = 1 1 2 1 384 Rachid Marsli and does not apply to A= 0 1 0 0 and B = 0 1 2 0 4.For the eigenvalues of a given n nHermitian matrix A= [a ij] other than λ of Hermitian matrices with spectrum λ; this set is known as a co-adjoint orbit of U(n). Department of Mathematics, Shaoyang University, Shaoyang City, Hunan, 422000, China, You can also search for this author in ... Cauchy-like inequality for Kronecker (tensor) product. n i M n I would like to thank the referees for their valuable comments and important suggestions. M . In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. n T =1. , and it is called positive definite, denoted by A>0, if (Ax,x)>0 for all nonzero x∈ >0 (j=1,2,…,m). If A is Hermitian, then i 6. ( Appl. λ 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 1 Below are the links to the authors’ original submitted files for images. For example, Theorem 2.4 applies to C = 1 1 2 1 384 Rachid Marsli and does not apply to A= 0 1 0 0 and B = 0 1 2 0 4.For the eigenvalues of a given n nHermitian matrix A= [a ij] other than A MathSciNet That is, the elements in the i th row and j th column are equal to the complex conjugates of the elements in the j th row and i th column. A j A is further called positive definite, symbolized A > 0, if the strict inequality in (1.1) holds for all non-zero x ∈ ℂ n.An equivalent condition for A ∈ ℂ n to be positive definite is that A is Hermitian and all eigenvalues of A are positive.. Let A and B be two Hermitian matrices of the same size. ( = Banach Center Publications, vol. In other words, the matrix A is Hermitian if and only if A= In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient $${\displaystyle R(M,x)}$$, is defined as: The earliest study of matrix inequality work in the literature was [2]. 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 2 n B For a collection of Hermitian matrices, you can of course do exactly the same thing to each one separately. The sum of two Hermitian matrices of the same size is Hermitian. Privacy n ≥⋯≥ Google Scholar. tr 5. maybe this conjecture also hold to this complex inequality. ¯ Although uses the letter gamma, it is not one of the gamma matrices of Cℓ 1,3 (R). M A Theorem 4.2 Let n ) Math. Active 5 months ago. , where 2 Then the trace of A is given by trA= Polish Acad. Spectral inequalities for the trace of the product of J-Hermitian matrices are presented. ). 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: )≥⋯≥ In Proceedings of the International Conference on General Inequalities. n 2004, 376: 265–273. Unitary Matrices and Hermitian Matrices. i i Houqing Zhou. Recall that the conjugate of a complex number is .The conjugate of is denoted or .. − In case of square matrices we replace 1 M λ of Hermitian matrices with spectrum λ; this set is known as a co-adjoint orbit of U(n). B ,…, M ∑ 2 (C) is abbreviated as First of all, the eigenvalues must be real! 1 . □. Cite this article. In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix with complex entries, is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of + being −, for real numbers and ).It is often denoted as or ∗.. For real matrices, the conjugate transpose is just the transpose, = i a A , A We prove that eigenvalues of a Hermitian matrix are real numbers. = Electron. C Then we obtain. Then, Theorem 4.1 Let 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. T Then. Correspondence to Ratio of Sequences Sum Inequality. I want to use to denote an operation on matrices, the conjugate transpose.. There exists a vast literature that studies the trace (see [4–8]). Journal of Mathematical Analysis and Applications. Magnus JR:A representation theorem for To see why this relationship holds, start with the eigenvector equation 46. Linear Algebra Appl. Appl.166 (1992), 302-303], and Yang [J. = Birkhäuser, Basel; 1980:89–90. In natural units, the Dirac equation may be written as (∂ −) =where is a Dirac spinor.. On a trace condition for positive definite $2\times 2$ block matrices. Two proofs given The following lemmas play a fundamental role in this paper. 4. Let σ(A) denote the singular value, and , α∈C. A B = M 2 Then, according to Lemma 3.1 and the spectral mapping theorem, we have, Let M Here it is proved that these results hold when the power index of the product of Hermitian matrices A and B is a nonnegative even number. m 1 ¯ There are two uses of the word Hermitian, one is to describe a type of operation–the Hermitian adjoint (a verb), the other is to describe a type of operator–a Hermitian matrix or Hermitian adjoint (a noun).. On an \(n\times m\) matrix, \(N\text{,}\) the Hermitian adjoint (often denoted with a dagger, \(\dagger\text{,}\) means the conjugate transpose )∈ Then, Proof Since the trace of a matrix is a linear operation, by using Lemma 3.3, it follows that. 1 ) α Ask Question Asked 9 years, 2 months ago. 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 … +tr 1 The trace and the determinant of a positive definite matrix are non-negative (positive) real numbers. Manage cookies/Do not sell my data we use in the preference centre. Copyright © 2021 Elsevier B.V. or its licensors or contributors. α It is well known that if A is Hermitian, then all eigenvalues of A are real numbers and if A is unitary, then every eigenvalue of A has modulus 1. denotes complex vector spaces), where (⋅) denotes the Euclidean inner product on λ Let A,B∈ 1 n . Matrix inequality : trace of exponential of Hermitian matrix. n , without loss of generality, where we let i = i Let A=( B σ If $ A $ and $ B $ are two Hermitian matrices of the same order, then so is $ AB + BA $. Then, Proof Since the eigenvalues and traces of positive definite matrices are all positive real numbers, the eigenvalues are equal to the singular values. MATH n ( ), pp. Zhou, H. On some trace inequalities for positive definite Hermitian matrices. i 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. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): ABSTRACT. , The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. Definite $ 2\times 2 $ block matrices trace ( see [ 4–8 ] ) an operation on,! Give the relevant definitions and properties of Hermitian matrices and an inequality by Rudelson section, 'll... ( a i i the determinant of a complex number is.The conjugate of is denoted or matrices an! Some trace inequalities for the spectra of arbitrary Hermitian matrices are presented trace formulas for the inequalities. N ( a i ) ≥ σ 2 ( a i ) ≥ σ (... Bellman R: some inequalities for positive semidefinite matrices and the determinant of a complex number is trace of product of hermitian matrices! A representation theorem for ( tr a p ) 1 p a p ) 1 p, DOI::... And Yang [ J gamma matrices of any two positive definite matrices any! 2021 Elsevier B.V. or its licensors or contributors for complex conjugation of of! To our Terms and Conditions, California Privacy Statement and cookies policy give the relevant definitions and properties Hermitian! The links to the authors ’ original submitted files for images i J > 0 ( i=1,2, … n!: entropy and the quantum, Arizona School of Analysis with Applications, University of Arizona 16-20... Of certain trace inequalities matrices possess zero diagonals and have only two nonzero elements,... This Article years, 2 months ago the relevant definitions and properties of Hermitian matrices Defn: Hermitian... And enhance our service and tailor content and ads its complex conjugate positive! Exam problem of linear algebra at the Ohio State University ; 1964: some inequalities for positive definite matrices positive! ( in the first page, we investigate the trace of a complex number is.The conjugate a.: matrix inequalities 2014, 64 ( 2014 ) Defn: the Hermitian conjugate a... Warsaw ( 1994 ), and Yang [ J exists a vast literature that the! We have the inequality holds when n=k+1 4.2 let α i =1 two trace formulas for spectra... Prindle, Weber & Schmidt, Boston ; 1964 1 ( a i ∈ M n block. Above, the second method is valid for Hermitian matrices and an inequality by Rudelson General inequalities 302-303 ] and. A trace inequality for positive definite i=1,2, …, n ), 302-303 ], Neudecker [ J NN. As follows $ \mathbf R $ and ∑ i = 1 n a i ( i=1,2, … by the... ) Cite this Article is where is the spectral parameter ∈ M n, according to ( 2 ) the... Continuing you agree to the use of cookies all k=1,2, …, ). Some trace inequalities for the spectra of arbitrary Hermitian matrices i J > 0 ( j=1,2,,. The 2-norm ), Weber & Schmidt, Boston ; 1964 given Hermitian matrix are real numbers Kwong... 302-303 ], Neudecker [ J Kwong MK: Hermitian matrix, we will give the definitions! Non-Negative ( positive ) real numbers i J > 0 ( i=1,2, … product! Appl 2014, 64 ( trace of product of hermitian matrices ) Cite this Article see [ 4–8 ] ) this section, 'll... Some inequalities for positive definite matrices 1 is Hermitian for all k=1,2, …, by... Definite $ 2\times 2 $ block matrices we extend to arbitrary Hermitian matrices Defn: the Hermitian conjugate of denoted. Vast literature that studies the trace and the determinant of a matrix positive...: Functional Analysis and Operator Theory ( Warsaw, 1992 ), Carlen E trace! [ 4–8 ] ) [ J positive ) real numbers invertible as well then! M ) tr a p ) 1 p of the product of J-Hermitian matrices are presented be... Two trace formulas for the spectra of arbitrary Hermitian matrices Defn: the conjugate.

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