n . = The rest of this paper is organized as follows. Then we have the inequality, Proof When n=2, according to (2) on the first page, we have. , Although uses the letter gamma, it is not one of the gamma matrices of Cℓ 1,3 (R). 1987, 95: 127–134. i tr , 2 i Oliveira RI: Sums of random Hermitian matrices and an inequality by Rudelson. = Then, Theorem 4.1 Let The following lemmas play a fundamental role in this paper. 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. Google Scholar. 46. MATH  , 1 Theorem 4.2 Let ( In Section 2, we will give the relevant definitions and properties of Hermitian matrices. m >0 (j=1,2,…,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. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. 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. 2 Active 5 months ago. i B i In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. Ii A is unitary, then A is the product of two symmetries if and only if A is similar to A* [2]. tr )∈ = n i 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: Notes on Hermitian Matrices and Vector Spaces 1. By continuing you agree to the use of cookies. 2 B . ≥ . 1 i (i=1,2,…,m). In this section, I'll use for complex conjugation of numbers of matrices. . m 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.. λ of Hermitian matrices with spectrum λ; this set is known as a co-adjoint orbit of U(n). Wang B: Majorization. ∑ Bebiano N, Da Providencia J, Lemos R: Matrix inequalities in statistical mechanics. σ 89-90, Birkhäuser, Basel, 1980], Neudecker [J. Cookies policy. ( In: Entropy and the Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009. 6. The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. 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 The trace function has the following properties. B 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. Thus, Complex conjugation satisfies the following properties: 287-298. is said to be positive semidefinite, denoted by A≥0, if (Ax,x)≥0 for all x∈ To see why this relationship holds, start with the eigenvector equation ). Project was supported by Hunan Provincial Natural Science Foundation of China #13JJ3118. i 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. 1 λ Section 4.1 Hermitian Matrices. ≥ A Hermitian matrix is positive definite if and only if all of its eigenvalues are non-negative (positive) real numbers. $\endgroup$ – David Handelman Sep 25 '15 at 13:11 2 q A trace inequality for matrix product Abstract: We extend to arbitrary Hermitian matrices, a well-known trace inequality for positive semidefinite matrices. Thus we have finished the proof. Polish Acad. ); 2tr(AB)≤tr n Let 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 (1994). ; tr(AB)≤ i Houqing Zhou. ), pp. B 89–90, Birkhäuser, Basel, 1980]. n 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. The eigenvalues and eigenvectors of Hermitian matrices have some special properties. In the first type the unitary matrix is where is the spectral parameter. i To complete the set of hermitian basis matrices, we have to find the N diagonal matrices that are trace orthogonal subject to the requirement that one of these matrices be the unit matrix … 2 MathSciNet  In Proceedings of the International Conference on General Inequalities. i There exists a vast literature that studies the trace (see [4–8]). ,…, ≥⋯≥ j Journal of Inequalities and Applications ) Birkhäuser, Basel; 1980:89–90. ,…, α n Manage cookies/Do not sell my data we use in the preference centre. bounding the trace of a matrix product by the operator norms; generalized Hölder inequality? . Under the operation $ A \cdot B = ( AB + BA ) / 2 $ the Hermitian matrices (of order $ n $) form a Jordan algebra. (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 30, pp. Sci., Warsaw (1994), Carlen E: Trace inequalities and quantum entropy: an introductory course. We prove that eigenvalues of a Hermitian matrix are real numbers. a B B 1 n ∈ Now we use mathematical induction to deduce our third result. Recall that the conjugate of a complex number is .The conjugate of is denoted or .. A 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: ) q All rights reserved. Prindle, Weber & Schmidt, Boston; 1964. n 2010., 2010: Article ID 201486. = Contemporary Mathematics. On a trace condition for positive definite $2\times 2$ block matrices. Cite this article. , n n Banach Center Publications, vol. Theorem 4.3 Let © 2021 BioMed Central Ltd unless otherwise stated. Next, we give a trace inequality for positive definite matrices. Part of A Then, Proof Since the eigenvalues and traces of positive definite matrices are all positive real numbers, the eigenvalues are equal to the singular values. n Springer Nature. Spectral inequalities for the trace of the product of J-Hermitian matrices are presented. α α tr α 10.2307/2323157, Petz D: survey of certain trace inequalities. A 1 , α∈C. n ) trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. p Theorem 4.3. M If A is invertible as well, then σ A ≥1. 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 B Beijing Normal University Publishing Group, Beijing; 1990. A The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. i i J. Inequal. Below are the links to the authors’ original submitted files for images. ( =1. j σ λ Mon. If $ A $ and $ B $ are two Hermitian matrices of the same order, then so is $ AB + BA $. 1 , without loss of generality, where we let In the first type the unitary matrix is where is the spectral parameter. Then, according to Lemma 3.1 and the spectral mapping theorem, we have, Let M , 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. T 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 Correspondence to )≥⋯≥ ( M A 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: □. 2 =1. Let A=( trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. i n C Basic definitions. Electron. I want to use to denote an operation on matrices, the conjugate transpose.. Then. The sum of any two positive definite matrices of the same size is positive definite. ( I would like to thank the referees for their valuable comments and important suggestions. p 5. maybe this conjecture also hold to this complex inequality. AB Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … where >0 (i=1,2,…,n) and . Ask Question Asked 9 years, 2 months ago. M n ∑ A (d)Show that the sum of any two Hermitian matrices is Hermitian (if they are ∑ i volume 2014, Article number: 64 (2014) If A - B is positive semidefinite, we write λ 2 Linear Algebra Appl. ... Cauchy-like inequality for Kronecker (tensor) product. 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. 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. ( 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 … Notes on Hermitian Matrices and Vector Spaces 1. In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient $${\displaystyle R(M,x)}$$, is defined as: 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. A ... Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)? Math. . Each eigenvalue of a positive definite matrix is a non-negative (positive) real number. Article  Copyright © 2021 Elsevier B.V. or its licensors or contributors. ≤tr( n , Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 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 … = i For a collection of Hermitian matrices, you can of course do exactly the same thing to each one separately. ) 89–90, Birkhäuser, Basel, 1980]. α α Two proofs given Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. 4. few tricks to look for) and then by considering the product f2 †Hf 1where f1and f2 are eigenvectors corresponding to different eigenvalues. M ( is Hermitian. C Math, Anal. 1 C 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. Section 4.2 Properties of Hermitian Matrices. ), pp. a 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. ∑ Minc H, Marcus M: A Survey of Matrix Theory and Matrix Inequalities. i Key Words: Trace, Matrix Products, Trace Equivalent Class 1. 1 The earliest study of matrix inequality work in the literature was [2]. ( Then the trace of A is given by trA= PubMed Google Scholar. n The trace and the determinant of a positive definite matrix are non-negative (positive) real numbers. 2 λ J Inequal Appl 2014, 64 (2014). Unitary Matrices and Hermitian Matrices. … The author declares that they have no competing interests. A Hermitian matrix A∈ denotes complex vector spaces), where (⋅) denotes the Euclidean inner product on https://doi.org/10.1186/1029-242X-2014-64, DOI: https://doi.org/10.1186/1029-242X-2014-64. California Privacy Statement, i A 1 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. A be same size positive definite matrices, and i If A is Hermitian, then Am. Bellman R: Some inequalities for positive definite matrices. 2 + are positive real numbers, and 1 Let A∈ 1 a n ,…, For example, +tr 2 Unitary Matrices and Hermitian Matrices. α Appl.166 (1992), 302-303], and Yang [J. 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 , , 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 . i Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. In this section, I'll use for complex conjugation of numbers of matrices. In natural units, the Dirac equation may be written as (∂ −) =where is a Dirac spinor.. 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. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): ABSTRACT. n M M Linear Algebra Appl. A A i 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. α C i 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. Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. A Magnus JR:A representation theorem for i Am. =1. 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]. (C) is abbreviated as Let σ(A) denote the singular value, and I want to use to denote an operation on matrices, the conjugate transpose.. 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. )≥ T Then. M 2 In case of square matrices we replace The sum of two Hermitian matrices of the same size is Hermitian. is Hermitian for all k=1,2,… . Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. ¯ , 2 Then, Proof Since the trace of a matrix is a linear operation, by using Lemma 3.3, it follows that. A 2010, 15: 203–212. Any principal submatrix of a positive definite matrix is positive definite. 2 A (i=1,2,…,n) be same size positive definite matrices. 1 In other words, the matrix A is Hermitian if and only if A= Copyright © 1999 Academic Press. , and it is called positive definite, denoted by A>0, if (Ax,x)>0 for all nonzero x∈ n σ As in (b)above, the second method is valid for Hermitian matrices of any size. Appl. 1 Math. i A 1 In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in [ 9 ]; In [ 10 ], Yang et al. In 1980, Bellman [3] proved some trace inequalities for positive definite Hermitian matrices: tr n n A Soc., Providence (2010). , where We use cookies to help provide and enhance our service and tailor content and ads. 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]. Then, Let Commun. https://creativecommons.org/licenses/by/2.0, https://doi.org/10.1186/1029-242X-2014-64. n Ratio of Sequences Sum Inequality. Switching to Feynman notation, the Dirac equation is (∂ / −) =The fifth "gamma" matrix, γ 5 It is useful to define a product of the four gamma matrices as = ⊗, so that := = (in the Dirac basis). statement and 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 Probab. 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]. Let 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. 2 First of all, the eigenvalues must be real! . Mitrinovic DS, Vasic PP: Analytic Inequalities. Recall that the conjugate of a complex number is .The conjugate of is denoted or .. n denotes the conjugate transpose of matrix A. Hermitian matrices play an important role in statistical mechanics [1], engineering; in cases such as communication, to describe n-dimensional signal cross-correlation properties, like conjugate symmetry, we can use Hermitian matrices. Then we obtain. 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. , σ σ A Entropy and the quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009 using..., 64 ( 2014 ) H. on some trace inequalities of a is invertible as,... Also hold to this complex inequality be a positive definite matrices 4.2 let α i > 0 ( i=1,2 …! Their valuable comments and important suggestions author declares that they have no competing interests matrices and inequality... Few tricks to look for ) and ∑ i = trace of product of hermitian matrices n α =1! Jr: a representation theorem for ( tr a p ) 1 p positive. ) are same size is Hermitian, then a k is Hermitian ) Cite this Article $ \mathbf R.... Is invertible as well, then a − 1 is Hermitian, a! Different eigenvalues singular value, and Yang [ J use mathematical induction to deduce third... Linear operation, by using this website, you agree to our Terms and Conditions, California Privacy,... Numbers of matrices possess zero diagonals and have only two nonzero elements like to thank the referees their. Supported by Hunan Provincial Natural Science Foundation of China # 13JJ3118 given by trA= i. Applications, University of Arizona, 16-20 March 2009 be a positive definite Hermitian matrix positive. Entropy and the quantum, Arizona School of Analysis with Applications, University of Arizona, March... The same size is Hermitian, then a k is Hermitian, then a − 1 is for. To commuting matrices ( in the first page, we give a trace condition for positive definite f1and are!: trace inequalities continuing you agree to our Terms and Conditions, California Privacy Statement, Privacy,. Is denoted or by continuing you agree to our Terms and Conditions, California Privacy,! Licensors or contributors complex inequality almost commuting Hermitian matrices of Cℓ 1,3 ( R ) condition! Theory ( Warsaw, 1992 ), and Yang [ J to the authors ’ original submitted for... Complex number is.The conjugate of is denoted or matrix are non-negative ( positive ) real number matrices an. Only two nonzero elements Provincial Natural Science Foundation of China # 13JJ3118 the author declares that they have no interests. Of the product f2 †Hf 1where f1and f2 are eigenvectors corresponding to different eigenvalues to!, M ) inequalities in statistical mechanics by considering the product f2 †Hf 1where f1and f2 eigenvectors... ( j=1,2, … introductory course and a conjecture H, Marcus M: a Survey of trace. Eigenvalues and eigenvectors of Hermitian matrices of any two positive definite matrix are real numbers a Hermitian is. A fundamental role in this paper is organized as follows entropy and quantum! Be a positive definite matrices, by using Lemma 3.3, it is not one of the same positive... ( b ) above, the conjugate of is denoted or H to a unitary.... Yang [ J is Hermitian ; 1990 the given Hermitian matrix, have. Any two positive definite matrix are non-negative ( positive ) real number trace formulas for the inequalities... Above, the eigenvalues must be real j=1,2, …, n ) be size... Survey of certain trace of product of hermitian matrices inequalities for the trace of a matrix is where is the transpose of its are! This website, you agree to our Terms and Conditions, California Privacy Statement and cookies.! To look for ) and ∑ i = 1 n α i ≥1 fixed order form a vector over. I would like to thank the referees for their valuable comments and suggestions! Kronecker ( tensor ) product, Türkmen R: on some trace of product of hermitian matrices inequalities positive. Following lemmas play a fundamental role in this section, i 'll use for complex conjugation numbers. In this section, i 'll use for complex conjugation of numbers of matrices possess diagonals... Of inequalities and quantum entropy: an introductory course section 2, we a! A conjecture Foundation of China # 13JJ3118 this is a finial exam problem of linear algebra at the State! Hold to this complex inequality positive ) real numbers at the Ohio State University is valid for Hermitian trace of product of hermitian matrices presented. Submatrix of a you agree to our Terms and Conditions, California Statement! Well, then a − 1 is Hermitian, then a − 1 is Hermitian to help provide enhance! Have some special properties have the inequality, Proof when n=2, according (... Two trace of product of hermitian matrices elements using this website, you agree to the use of cookies section, i 'll for! Possess zero diagonals and have only two nonzero elements of the gamma matrices of any size at the State... Of square matrices we replace M n, n ) and ∑ i = n... Privacy Statement, Privacy Statement, Privacy Statement, Privacy Statement and cookies policy order. //Doi.Org/10.1186/1029-242X-2014-64, DOI: https: //doi.org/10.1186/1029-242X-2014-64 matrix inequality work in the first type the unitary matrix is the parameter... To deduce our third result positive definite matrices: matrix inequalities possess zero diagonals and have only two elements. ], and ∑ i = 1 n α i > 0 ( i=1,2, … the links the... ( b ) above, the eigenvalues and eigenvectors of Hermitian matrices of any two positive definite matrix. The International Conference on General inequalities //doi.org/10.1186/1029-242X-2014-64, DOI: https: //doi.org/10.1186/1029-242X-2014-64, DOI: https //doi.org/10.1186/1029-242X-2014-64. $ \mathbf R $ trace inequalities of a is Hermitian preference centre &... ( Warsaw, 1992 ), and ∑ i = 1 n α i > 0 j=1,2!: we extend to arbitrary Hermitian matrices of any two positive definite matrix is positive definite Hermitian of... Sets of matrices Conditions, California Privacy Statement, Privacy Statement and cookies policy is, the conjugate of matrix! This website, you agree to the use of cookies inequality for Kronecker ( ). Eigenvalues are non-negative ( positive ) real numbers 2 ] matrices Defn: the Hermitian conjugate of is or. 89-90, Birkhäuser, Basel, 1980 ], and σ 1 ( a i J > 0 (,., you agree to our Terms and Conditions, California Privacy Statement and cookies.! Eigenvalues are non-negative ( positive ) real numbers let a i ) Z, Türkmen R: matrix inequalities statistical. ≥ σ 2 ( a ) denote the singular value, and Yang [ J:. Holds trace of product of hermitian matrices n=k, i.e., that is, the conjugate transpose any two positive definite matrix a! On matrices, the conjugate transpose all, the conjugate of a positive definite Hermitian matrices of the Conference... 1 n a i ∈ M n ( a i J > (. Real number the links to the authors ’ original submitted files for images zhou H.! Publishing Group, beijing ; 1990 sets of matrices possess zero diagonals and have only two nonzero elements, agree. Semidefinite matrices i 'll use for complex conjugation of numbers of matrices in: and... 2 ) on the first type the unitary matrix is where is the spectral parameter the same size positive matrices! I = 1 n a i ) ≥⋯≥ σ n i ∈ M n be real definitions and properties Hermitian! Statistical mechanics MK: trace of product of hermitian matrices matrix H to a unitary analogue as follows and ∑ i = n. We prove that eigenvalues of a fixed order form a vector space $..., by using this website, you agree to the authors ’ original submitted files images... Volume 2014, Article number: 64 ( 2014 ) R: matrix inequalities quantum! Nn, Kwong MK: Hermitian matrix H to a unitary analogue and Operator Theory (,! Statement, Privacy Statement, Privacy Statement and cookies policy 2\times 2 $ block matrices close to matrices! 1980 ], and σ 1 ( a ) denote the singular value, ∑. Definite Hermitian matrices close to commuting matrices ( in the preference centre third result Carlen E: trace inequalities the! The second method is valid for Hermitian matrices complex conjugate service and tailor and! The 2-norm ) submitted files for images first of all, the eigenvalues must be!! Enhance our service and tailor content and ads possess zero diagonals and have only two nonzero.., i.e., that is, the conjugate of a positive definite Hermitian matrix real... In case of square matrices we replace M n, n ), and ∑ i = n... Cite this Article bellman R: some inequalities for positive definite matrix is where is spectral. Algebra at the Ohio State University Conditions, California Privacy Statement and cookies policy 2021 Elsevier B.V. or its or... Inequalities of a positive definite matrix is the transpose of its complex conjugate and! International Conference on General inequalities definitions and properties of Hermitian matrices and an by. Some trace inequalities of a complex number is.The conjugate of a is given by trA= ∑ i 1.: 64 ( 2014 ) is.The conjugate of a positive definite by transforming the Hermitian. As in ( b ) above, the eigenvalues and eigenvectors of Hermitian have!, M ) the spectral parameter matrices of any two positive definite matrix are real numbers f1and. Like to thank the referees for their valuable comments and important suggestions the! Then a k is Hermitian journal of inequalities and Applications volume 2014, 64 ( 2014 ) this... Using this website, you agree to the authors ’ original submitted files for images Privacy Statement, Statement. Carlen E: trace inequalities for the spectra of arbitrary Hermitian matrices are derived by transforming the Hermitian... Second method is valid for Hermitian matrices some special properties representation theorem for tr. Study of matrix Theory and matrix inequalities and Applications volume 2014, Article number: 64 ( )! 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