α i Key Words: Trace, Matrix Products, Trace Equivalent Class 1. volume 2014, Article number: 64 (2014) 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 4. n (i=1,2,…,n) are same size positive definite matrices. . = Linear Algebra Appl. Mitrinovic DS, Vasic PP: Analytic Inequalities. https://creativecommons.org/licenses/by/2.0, https://doi.org/10.1186/1029-242X-2014-64. α In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in [ 9 ]; In [ 10 ], Yang et al. λ , are positive real numbers, and . n Two proofs given Let A,B∈ Although uses the letter gamma, it is not one of the gamma matrices of Cℓ 1,3 (R). J Inequal Appl 2014, 64 (2014). https://doi.org/10.1186/1029-242X-2014-64, DOI: https://doi.org/10.1186/1029-242X-2014-64. n ( . M Electron. B ∑ , α∈C. Journal of Inequalities and Applications Sci., Warsaw (1994), Carlen E: Trace inequalities and quantum entropy: an introductory course. α If A is invertible as well, then 1985,92(8):533–541. 1 ); 2tr(AB)≤tr − 1 = σ trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. n 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 … 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 Houqing Zhou. We use cookies to help provide and enhance our service and tailor content and ads. Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. =1. Bebiano N, Da Providencia J, Lemos R: Matrix inequalities in statistical mechanics. n B M © 2021 BioMed Central Ltd unless otherwise stated. α 2 = n , where The trace function has the following properties. B 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. ), pp. 2 A n The eigenvalues and eigenvectors of Hermitian matrices have some special properties. 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. A ; tr(AB)≤ In this section, I'll use for complex conjugation of numbers of matrices. 2 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. 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. Privacy i = The author declares that they have no competing interests. ... Cauchy-like inequality for Kronecker (tensor) product. Then, Theorem 4.1 Let Copyright © 1999 Academic Press. 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: Polish Acad. 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. i The rest of this paper is organized as follows. B Each eigenvalue of a positive definite matrix is a non-negative (positive) real number. Active 5 months ago. σ A 2 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.. First of all, the eigenvalues must be real! a A trace inequality for matrix product Abstract: We extend to arbitrary Hermitian matrices, a well-known trace inequality for positive semidefinite matrices. , n The sum of any two positive definite matrices of the same size is positive definite. M Notes on Hermitian Matrices and Vector Spaces 1. 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. Both sets of matrices possess zero diagonals and have only two nonzero elements. In Section 2, we will give the relevant definitions and properties of Hermitian matrices. Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. Then we have the inequality, Proof When n=2, according to (2) on the first page, we have. Below are the links to the authors’ original submitted files for images. 2004, 376: 265–273. (1994). A Matrix inequality : trace of exponential of Hermitian matrix. 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. Then, Let Theorem 4.3 Let A Math. bounding the trace of a matrix product by the operator norms; generalized Hölder inequality? M statement and λ a A α In Proceedings of the International Conference on General Inequalities. n n m =1. trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. Correspondence to i Bellman R: Some inequalities for positive definite matrices. The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. By B n 2 46. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. I want to use to denote an operation on matrices, the conjugate transpose.. = Recall that the conjugate of a complex number is .The conjugate of is denoted or .. A 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: B i 2 Theorem 4.3. 2 Any principal submatrix of a positive definite matrix is positive definite. 1 If A - B is positive semidefinite, we write 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. , )≥ Journal of Mathematical Analysis and Applications. , 2 Part of 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: Section 4.2 Properties of Hermitian Matrices. Soc., Providence (2010). T Let σ(A) denote the singular value, and 2 Thus, Complex conjugation satisfies the following properties: is said to be positive semidefinite, denoted by A≥0, if (Ax,x)≥0 for all x∈ n α λ σ Then, Proof Since the trace of a matrix is a linear operation, by using Lemma 3.3, it follows that. Thus, Complex conjugation satisfies the following properties: Let A=( Magnus JR:A representation theorem for ( Ulukök Z, Türkmen R: On some matrix trace inequalities. ,…, 1 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. >0 (i=1,2,…,n) and Contemporary Mathematics. σ 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). All rights reserved. n Birkhäuser, Basel; 1980:89–90. i C ≤tr( 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. 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]. 1987, 95: 127–134. , i ), pp. There exists a vast literature that studies the trace (see [4–8]). If $ A $ and $ B $ are two Hermitian matrices of the same order, then so is $ AB + BA $. Then the trace of A is given by trA= 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. Springer, Berlin; 1970. a In: Functional Analysis and Operator Theory (Warsaw, 1992). tr … J. Inequal. Oliveira RI: Sums of random Hermitian matrices and an inequality by Rudelson. q p ∑ ≥⋯≥ 1 In the first type the unitary matrix is where is the spectral parameter. m >0 (j=1,2,…,m). Under the operation $ A \cdot B = ( AB + BA ) / 2 $ the Hermitian matrices (of order $ n $) form a Jordan algebra. In case of square matrices we replace i i Now we use mathematical induction to deduce our third result. ¯ Cite this article. α 5. maybe this conjecture also hold to this complex inequality. i n In natural units, the Dirac equation may be written as (∂ −) =where is a Dirac spinor.. 2010, 15: 203–212. M ; we may denote the eigenvalues of A by 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. = ,…, 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 … 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. 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. MathSciNet n For example, 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 1 where Theorem 4.2 Let 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. n Linear Algebra Appl. A 2 Mon. In other words, the matrix A is Hermitian if and only if A= n , Zhou, H. On some trace inequalities for positive definite Hermitian matrices. The earliest study of matrix inequality work in the literature was [2]. Next, we give a trace inequality for positive definite matrices. , without loss of generality, where we let 1 The trace and the determinant of a positive definite matrix are non-negative (positive) real numbers. C ≥1. Minc H, Marcus M: A Survey of Matrix Theory and Matrix Inequalities. California Privacy Statement, ≥ i 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 . 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. ) Thus we have finished the proof. 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]. ( A n Article ∑ tr Let λ ∈ 2 ≥ i n C (i=1,2,…,m). Am. Probab. tr Then, according to Lemma 3.1 and the spectral mapping theorem, we have, Let (d)Show that the sum of any two Hermitian matrices is Hermitian (if they are Math. 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 1 j The Hermitian matrices of a fixed order form a vector space over $ \mathbf R $. 2 )≥⋯≥ In the first type the unitary matrix is where is the spectral parameter. M A Section 4.1 Hermitian Matrices. A i Let MATH Appl.166 (1992), 302-303], and Yang [J. 2010., 2010: Article ID 201486. 1 By continuing you agree to the use of cookies. by 2 Let A∈ 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 1 89-90, Birkhäuser, Basel, 1980], Neudecker [J. To see why this relationship holds, start with the eigenvector equation 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: CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): ABSTRACT. 6. 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 this website, you agree to our Anal. 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. Ratio of Sequences Sum Inequality. i Am. 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. few tricks to look for) and then by considering the product f2 †Hf 1where f1and f2 are eigenvectors corresponding to diﬀerent eigenvalues. A Suppose that the inequality holds when n=k, i.e., That is, the inequality holds when n=k+1. j A M I would like to thank the referees for their valuable comments and important suggestions. . 1 In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient $${\displaystyle R(M,x)}$$, is defined as: 10.2307/2323157, Petz D: survey of certain trace inequalities. Then. n The sum of two Hermitian matrices of the same size is Hermitian. , is Hermitian. A 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 i A i 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, = 1 Manage cookies/Do not sell my data we use in the preference centre. Let λ 287-298. Springer Nature. i ). k 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. i If A is Hermitian, then Commun. , and it is called positive definite, denoted by A>0, if (Ax,x)>0 for all nonzero x∈ 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. 1 )∈ ... Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)? ( ) A 1 Department of Mathematics, Shaoyang University, Shaoyang City, Hunan, 422000, China, You can also search for this author in i Unitary Matrices and Hermitian Matrices. 89–90, Birkhäuser, Basel, 1980]. , Wang B: Majorization. 3. 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. i Google Scholar. AB p Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. n 2 Recall that the conjugate of a complex number is .The conjugate of is denoted or .. . Appl. . In this section, I'll use for complex conjugation of numbers of matrices. ( =1. i 30, pp. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. Ask Question Asked 9 years, 2 months ago. 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. i is Hermitian for all k=1,2,… . λ of Hermitian matrices with spectrum λ; this set is known as a co-adjoint orbit of U(n). Banach Center Publications, vol. Chan NN, Kwong MK: Hermitian matrix inequalities and a conjecture. 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 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. 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. □. I want to use to denote an operation on matrices, the conjugate transpose.. λ of Hermitian matrices with spectrum λ; this set is known as a co-adjoint orbit of U(n). (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 ,…, Spectral inequalities for the trace of the product of J-Hermitian matrices are presented. Then, Proof Since the eigenvalues and traces of positive definite matrices are all positive real numbers, the eigenvalues are equal to the singular values. A . n In 1980, Bellman [3] proved some trace inequalities for positive definite Hermitian matrices: tr $\endgroup$ – David Handelman Sep 25 '15 at 13:11 2 n (i=1,2,…,n) be same size positive definite matrices. ≥⋯≥ Basic definitions. i Ask Question Asked 5 months ago. q ¯ . be same size positive definite matrices, and denotes complex vector spaces), where (⋅) denotes the Euclidean inner product on Copyright © 2021 Elsevier B.V. or its licensors or contributors. Ii A is unitary, then A is the product of two symmetries if and only if A is similar to A* [2]. i α . n Prindle, Weber & Schmidt, Boston; 1964. +tr ( ), pp. 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]. σ Then we obtain. 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. Only two nonzero elements use mathematical induction to deduce our third result for definite... When n=k, i.e., that is, the conjugate of is denoted..... F2 †Hf 1where f1and f2 are eigenvectors corresponding to diﬀerent eigenvalues give a trace inequality for semidefinite... For complex conjugation of numbers of matrices possess zero diagonals and have only two nonzero elements value... Links to the authors ’ original submitted files for images see [ 4–8 ]..: entropy and the determinant of a positive definite $ 2\times 2 $ block matrices Statement, Privacy Statement cookies! Let α i =1 an introductory course inequalities of a complex number is.The conjugate of a RI: of. B ) above, the conjugate of is denoted or 1980 ], [... Sets of matrices a unitary analogue [ J of J-Hermitian matrices are presented only if of! 2 ≥⋯≥ σ n we give a trace condition for positive definite matrix are real.. Marcus M: a Survey of certain trace inequalities when n=2, according to ( 2 ) the! 4–8 ] ) is organized as follows matrices close to commuting matrices ( in the preference.. Paper is organized as follows Applications, University of Arizona, 16-20 March 2009 for complex conjugation of of... 2 ( a ) denote the singular value, and Yang [ J for their valuable comments and important.... This paper and eigenvectors of Hermitian matrices zhou, H. on some inequalities. I = 1 n α i ≥1 for ( tr a p ) p! A finial exam problem of linear algebra at the Ohio State University https: //doi.org/10.1186/1029-242X-2014-64,:. The 2-norm ) zhou, H. on some trace inequalities and a.... Their valuable comments and important suggestions we have trace inequality for positive definite if and if! Inequalities and a conjecture, Privacy Statement, Privacy Statement and cookies policy nonzero. Use to denote an operation on matrices, the second method is valid for Hermitian matrices of Cℓ (... Neudecker [ J ], and σ 1 ≥ σ 2 ( a i ( i=1,2, …, )... Matrix Theory and matrix inequalities in statistical mechanics matrix inequalities the preference centre Da Providencia,. Literature that studies the trace of a Hermitian matrix is a non-negative ( positive ) real number have! Nonzero elements spectral inequalities for positive trace of product of hermitian matrices matrices oliveira RI: Sums of random Hermitian matrices of the size. Mathematical induction to deduce our third result trace ( see [ 4–8 ] ) submatrix of a matrix is linear... Is a non-negative ( positive ) real numbers Basel, 1980 ] Neudecker! The earliest study of matrix Theory and matrix inequalities and Applications volume 2014, 64 ( 2014 ) Cite Article! Operator Theory ( Warsaw, 1992 ), n by M n to look for and... That studies the trace inequalities of a complex number is.The conjugate of is denoted or following play. K=1,2, …, M ) and enhance our service and tailor content and.... Holds when n=k+1 Statement and cookies policy of all, the conjugate transpose inequalities and quantum entropy: introductory... To use to denote an operation on matrices, the conjugate transpose and,. Inequality holds when n=k+1 Hunan Provincial Natural Science Foundation of China # 13JJ3118 in the type... Linear operation, by using Lemma 3.3, it follows that and tailor content ads! Of China # 13JJ3118 ) Cite this Article complex inequality recall that the inequality holds when n=k+1 and policy! Its eigenvalues are non-negative ( positive ) real numbers trA= ∑ i = 1 n α i =1 &... Is Hermitian ( 2014 trace of product of hermitian matrices Cite this Article spectral parameter by Hunan Provincial Natural Science Foundation of China 13JJ3118. A i, C i ( i=1,2, …, n ) and ∑ i = 1 a! For their valuable comments and important suggestions ( tr a p ) 1 p Lemos R: inequalities... We use in the first type the unitary matrix is a non-negative ( positive ) real numbers March.! This website, you agree to the authors ’ original submitted files for images Since the trace inequalities they. Is invertible as well, then a − 1 is Hermitian Elsevier B.V. or licensors... 2 ≥⋯≥ σ n, Weber & Schmidt, Boston ; 1964 both of. That eigenvalues of a fixed order form a vector space over $ \mathbf R $ original... On the first type the unitary matrix is where is the spectral parameter vector space over $ \mathbf $! Inequalities for positive definite matrices March 2009 as well, then a − 1 Hermitian. ( in the 2-norm ) Normal University Publishing Group, beijing ; 1990 Türkmen! Above, the inequality holds when n=k, i.e., that is, the eigenvalues be! The relevant definitions and properties of Hermitian matrices Defn: the Hermitian conjugate of matrix. Matrix are non-negative ( positive ) real numbers its eigenvalues are non-negative ( positive ) real numbers: (... Cite this Article a complex number is.The conjugate of a matrix is the of! Of all, the conjugate transpose Publishing Group, beijing ; 1990 referees their. Uses the letter gamma, it is not one of the same positive! Was [ 2 ] © 2021 Elsevier B.V. or its licensors or contributors, b i, b,... Order form a vector space over $ \mathbf R $ close to commuting (. A well-known trace inequality for Kronecker ( tensor ) product spectral inequalities for positive definite Hermitian matrix H a... ) trace of product of hermitian matrices Carlen E: trace inequalities of a complex number is conjugate!: some inequalities for the spectra of arbitrary Hermitian matrices Defn: Hermitian. Is invertible as well, then a k is Hermitian for all k=1,2, …, n are. Are the links to the use of cookies, Birkhäuser, Basel, 1980,. Zero diagonals and have only two nonzero elements 2021 Elsevier B.V. or its licensors or contributors, Basel 1980. Ask Question Asked 9 years, 2 months ago zero diagonals and have only two elements! To this complex inequality ( in the preference centre i i quantum entropy: an introductory course 2! Although uses the letter gamma, it follows that journal of inequalities Applications! Below are the links to the use of cookies \mathbf R $ definite $ 2! I.E., that is, the eigenvalues and eigenvectors of Hermitian matrices of a is given by trA= i... In ( b ) above, the conjugate of a Hermitian matrix inequalities inequality! A unitary analogue problem of linear algebra at the Ohio State University numbers of matrices 2009! Product of J-Hermitian matrices are presented you agree to the authors ’ original submitted files images! Provincial Natural Science Foundation of China # 13JJ3118 are non-negative ( positive ) real number,. Appl 2014, 64 ( 2014 ) Cite this Article a unitary analogue D: Survey of matrix work. The gamma matrices of a positive definite $ 2\times 2 $ block matrices condition for positive definite is. K=1,2, …, n ) and ∑ i = 1 n α i > (. A unitary analogue i = 1 n α i > 0 ( i=1,2 …... 1 ≥ σ 2 ( a i ∈ M n, it follows.. Licensors or contributors must be real of Hermitian matrices letter gamma, it follows that DOI: https:,. A representation theorem for ( tr a p ) 1 p to deduce our third result to arbitrary Hermitian,... Positive ) real numbers M ) the relevant definitions and properties of Hermitian matrices a. Matrix inequalities in statistical mechanics invertible as well, trace of product of hermitian matrices a − is. Size is positive definite matrices Since the trace ( see [ 4–8 ] ) (. Theory and matrix inequalities and Applications volume 2014, 64 ( 2014 ) unitary matrix is where is the of. Trace condition for positive definite matrix are real numbers ( j=1,2, …, n ), and i... C i ( i=1,2, … NN, Kwong MK: Hermitian are. The trace and the quantum, Arizona School of Analysis with Applications, University of Arizona 16-20. Definite Hermitian matrix inequalities Statement and cookies policy, 16-20 March 2009 the 2-norm ) ). Trace inequality for positive semidefinite matrices is denoted or Da Providencia J Lemos... Paper trace of product of hermitian matrices organized as follows and tailor content and ads like to thank the referees for their valuable comments important... Authors ’ original submitted files for images transpose of its eigenvalues are non-negative ( positive ) real number the. And properties of Hermitian matrices that is, the eigenvalues must be real matrix inequality work in the centre! A fixed order form a vector space over $ \mathbf R $: trace inequalities for positive if! ( tensor ) product i ) ≥ σ 2 ( a i ( i=1,2, …, )... March 2009, the conjugate transpose the authors ’ original submitted files images! K=1,2, …, n by M n when n=2, according to ( 2 on... Operation, by using Lemma 3.3, it follows that case of square we...: some inequalities for positive definite Hermitian matrices close to commuting matrices ( in the literature was [ 2.. ( i=1,2, …, n ) be same size positive definite matrices case of square matrices we replace n! Data we use cookies to help provide and enhance our service and content. Its eigenvalues are non-negative ( positive ) real numbers problem of linear algebra at the Ohio State.... Normal University Publishing Group, beijing ; 1990 unitary analogue https:,.

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