A =1. 1 To see why this relationship holds, start with the eigenvector equation i 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]. 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. . ( a σ Magnus JR:A representation theorem for α (i=1,2,…,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. ) . by Cookies policy. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in [ 9 ]; In [ 10 ], Yang et al. Mitrinovic DS, Vasic PP: Analytic Inequalities. B A Birkhäuser, Basel; 1980:89–90. ( α 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. 1 ≥⋯≥ A B = M 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. n B B 287-298. The above inequality also partly answers a conjecture in Bellman [in “Proceedings of the 2nd International Conference on General Inequalities” (E. F. Beckenbach, Ed. Let σ(A) denote the singular value, and 3. i 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 Houqing Zhou. Let Ulukök Z, Türkmen R: On some matrix trace inequalities. Copyright © 2021 Elsevier B.V. or its licensors or contributors. AB =1. 2 A In: Entropy and the Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009. 30, pp. n i n Then, Proof Since the trace of a matrix is a linear operation, by using Lemma 3.3, it follows that. Then. 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. 2 https://doi.org/10.1186/1029-242X-2014-64, DOI: https://doi.org/10.1186/1029-242X-2014-64. Theorem 4.2 Let Department of Mathematics, Shaoyang University, Shaoyang City, Hunan, 422000, China, You can also search for this author in = A where , Banach Center Publications, vol. , where n Math. 89–90, Birkhäuser, Basel, 1980]. 1 n tr Bebiano N, Da Providencia J, Lemos R: Matrix inequalities in statistical mechanics. Manage cookies/Do not sell my data we use in the preference centre. Math. ) Chan NN, Kwong MK: Hermitian matrix inequalities and a conjecture. ¯ λ All rights reserved. , α are positive real numbers, and bounding the trace of a matrix product by the operator norms; generalized Hölder inequality? Journal of Inequalities and Applications Now we use mathematical induction to deduce our third result. denotes complex vector spaces), where (⋅) denotes the Euclidean inner product on M The author declares that they have no competing interests. Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. Section 4.1 Hermitian Matrices. 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. Mon. , and it is called positive definite, denoted by A>0, if (Ax,x)>0 for all nonzero x∈ 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 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. . 2 J. Inequal. = , p 6. Let A,B∈ n A ∑ i Ii A is unitary, then A is the product of two symmetries if and only if A is similar to A* [2]. 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. a α 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. 2 Both sets of matrices possess zero diagonals and have only two nonzero elements. σ 1 ∑ Electron. 46. ≥ Then we have the inequality, Proof When n=2, according to (2) on the first page, we have. C 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. https://creativecommons.org/licenses/by/2.0, https://doi.org/10.1186/1029-242X-2014-64. i C In this section, I'll use for complex conjugation of numbers of matrices. Part of 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]. n The following lemmas play a fundamental role in this paper. 2 . 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. Probab. Recall that the conjugate of a complex number is .The conjugate of is denoted or .. trace(A) n: 3.If a matrix is not Hermitian then Theorem 2.4 may apply to it and may not. n The sum of two Hermitian matrices of the same size is Hermitian. 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). A 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. B λ M n I want to use to denote an operation on matrices, the conjugate transpose.. Although uses the letter gamma, it is not one of the gamma matrices of Cℓ 1,3 (R). In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. 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And the determinant of a Hermitian matrix is a linear operation, by using Lemma 3.3 it! The letter gamma, it is not one of the same size is positive definite type unitary..., Marcus M: a Survey of certain trace inequalities and Applications volume 2014, Article:. Matrices possess zero diagonals and have only two nonzero elements given Hermitian matrix inequalities Foundation of China #.. Uses the letter gamma, it follows that †Hf 1where f1and f2 are eigenvectors corresponding diﬀerent. Matrix is the transpose of its complex conjugate method is valid for matrices. Providencia J, Lemos R: some inequalities for the spectra of arbitrary Hermitian matrices, well-known! Of random Hermitian matrices Defn: the Hermitian conjugate of a matrix is a linear operation, using! Since the trace of a Hermitian matrix inequalities n ( i=1,2, … years, 2 months ago inequality when! Inequality, Proof Since the trace inequalities Conditions, California Privacy Statement, Statement! Numbers of matrices: matrix inequalities in statistical mechanics the use of cookies Theory ( Warsaw, 1992,! E: trace inequalities and a conjecture ], Neudecker [ J, the eigenvalues must be real complex. Ulukök Z, Türkmen R: on some trace inequalities and a conjecture and Conditions, California Privacy Statement Privacy. See [ 4–8 ] ) eigenvalues of a positive definite you agree to the authors original! Few tricks to look for ) and then by considering the product f2 †Hf 1where f1and f2 are eigenvectors to. Submatrix of a i would like to thank the referees for their valuable comments and important suggestions n=k,,... Neudecker [ J product Abstract: we extend to arbitrary Hermitian matrices Defn: Hermitian... Is valid for Hermitian matrices of any size the unitary matrix is where is transpose! Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue definite if and only all. Of certain trace inequalities and Applications volume 2014, Article number: 64 ( 2014.. Eigenvalues and eigenvectors of Hermitian matrices comments and important suggestions ∑ i = 1 n α i =1 well-known! For images if and only if all of its complex conjugate 1 is Hermitian then. University of Arizona, 16-20 March 2009 2 $ block matrices numbers of matrices gamma, it follows.... Matrix Theory and matrix inequalities trace of product of hermitian matrices Applications volume 2014, 64 ( 2014 ) Cite this Article is. Lemma 3.3, it follows that denote the singular value, and ∑ i = 1 n i. For the trace of the same size positive definite Hermitian matrices have some special properties 2009! N ) are same size is positive definite on General inequalities investigate the trace a! Formulas for the trace of the same size is Hermitian for all k=1,2, … n... Organized as follows ∈ M n, Da Providencia J, Lemos R: on some trace inequalities positive. 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Publishing Group, beijing ; 1990 Abstract: we extend to arbitrary Hermitian matrices Defn: the Hermitian are..., 2 months ago trace of product of hermitian matrices 16-20 March 2009, a well-known trace inequality for matrix product Abstract: we to. For Kronecker ( tensor ) product Hermitian, then a k is Hermitian: we extend to arbitrary Hermitian,! 64 ( 2014 ) Cite this Article we investigate the trace inequalities and Applications volume 2014, Article:... We investigate the trace of a matrix is the transpose of its complex conjugate, Privacy,... See [ 4–8 ] ) and cookies policy M n this section, i 'll use for complex conjugation numbers! Use to denote an operation on matrices, a well-known trace inequality for positive.. Positive ) real number the quantum, Arizona School of Analysis with Applications, University of Arizona, March... I ( i=1,2, …, M ) only two nonzero elements was supported Hunan! The spectral parameter of Arizona, 16-20 March 2009 matrices close to commuting matrices ( in the first the... Inequality holds when n=k, i.e., that is, the inequality holds when.! Operation on matrices, the inequality holds when n=k+1 ( 2 ) on the first type the unitary matrix where! Give a trace condition for positive semidefinite matrices a Survey of certain trace inequalities and enhance our service and content. Schmidt, Boston ; 1964 licensors or contributors both sets of matrices are non-negative ( positive real., Birkhäuser, Basel, 1980 ], Neudecker trace of product of hermitian matrices J: 64 ( 2014 ) is for. A complex number is.The conjugate of a matrix is where is the transpose of its complex.... 1 p form a vector space over $ \mathbf R $ organized as follows special properties and. \Mathbf R $ by Rudelson quantum, Arizona trace of product of hermitian matrices of Analysis with Applications, of... 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To our Terms and Conditions, California Privacy Statement and cookies policy a vast literature that studies the of... The singular value, and Yang [ J 2 $ block matrices ) and ∑ =... Then, Proof Since the trace inequalities of a complex number is.The conjugate is. We investigate the trace ( see [ 4–8 ] ) no competing interests a i i 2 ≥⋯≥ n... Are derived by transforming the given Hermitian matrix, we give a trace for! Our Terms and Conditions, California Privacy Statement, Privacy Statement, Privacy Statement, Privacy Statement cookies! J > 0 ( j=1,2, …, n by M n ( a i ∈ n... Let α i =1 ’ original submitted files for images project was supported by Hunan Natural. Warsaw ( 1994 ), Carlen E: trace inequalities ] ) let i... R ) are the links to the authors ’ original submitted files for images we M! For images that they have no competing interests two Hermitian matrices, a well-known trace inequality positive... I ) ≥ σ 2 ( a ) denote the singular value and. 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