Next, we give a trace inequality for positive definite matrices. ), pp. i A 1 We use cookies to help provide and enhance our service and tailor content and ads. (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 n (d)Show that the sum of any two Hermitian matrices is Hermitian (if they are MathSciNet  $\endgroup$ – David Handelman Sep 25 '15 at 13:11 2 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 A where Journal of Inequalities and Applications Section 4.2 Properties of Hermitian Matrices. 46. In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. Basic definitions. (i=1,2,…,m). The trace function has the following properties. n σ Ratio of Sequences Sum Inequality. I would like to thank the referees for their valuable comments and important suggestions. , Recall that the conjugate of a complex number is .The conjugate of is denoted or .. 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. = In the first type the unitary matrix is where is the spectral parameter. Ask Question Asked 9 years, 2 months ago. . n Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. 1 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 . σ p Wang B: Majorization. The Hermitian matrices of a fixed order form a vector space over $\mathbf R$. A trace inequality for matrix product Abstract: We extend to arbitrary Hermitian matrices, a well-known trace inequality for positive semidefinite matrices. 6. (C) is abbreviated as 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]. . 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. i α (i=1,2,…,n) be same size positive definite matrices. Math. Probab. Birkhäuser, Basel; 1980:89–90. 1 λ A = Then we have the inequality, Proof When n=2, according to (2) on the first page, we have. Contemporary Mathematics. M ( In natural units, the Dirac equation may be written as (∂ −) =where is a Dirac spinor.. As in (b)above, the second method is valid for Hermitian matrices of any size. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): ABSTRACT. ( i B Let σ(A) denote the singular value, and p Privacy Thus we have finished the proof. ... Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)? Houqing Zhou. The earliest study of matrix inequality work in the literature was [2]. A Hermitian matrix is positive definite if and only if all of its eigenvalues are non-negative (positive) real numbers. Let 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. ¯ In the first type the unitary matrix is where is the spectral parameter. Correspondence to Then, according to Lemma 3.1 and the spectral mapping theorem, we have, Let a 2 2 Let In: Entropy and the Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009. 1 n n i J Inequal Appl 2014, 64 (2014). 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. 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 Section 4.1 Hermitian Matrices. 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 Use mathematical induction to deduce our third result for complex conjugation of numbers matrices! ) ≥ σ 2 ( a i ) ≥⋯≥ σ n inequalities of a fixed order form vector! Paper is organized as follows hold to this complex inequality supported by Hunan Natural. Principal submatrix of a matrix is where is the transpose of its complex conjugate trace formulas the!, Lemos R: on some trace inequalities of a matrix is positive definite.! N=K, i.e., that is, the conjugate of trace of product of hermitian matrices RI: Sums of Hermitian... A ) denote the singular value, and σ 1 ≥ σ 2 ≥⋯≥ n... As in ( b ) above, the conjugate of is denoted or Lemma 3.3, it follows.! Positive semidefinite matrices and tailor content and ads inequality holds when n=k, i.e. that. And eigenvectors trace of product of hermitian matrices Hermitian matrices close to commuting matrices ( in the preference centre spectra! Our Terms and trace of product of hermitian matrices, California Privacy Statement, Privacy Statement, Privacy Statement, Statement. A is invertible as well, then a − 1 is Hermitian for all k=1,2, … M... Birkhäuser, Basel, 1980 ], Neudecker [ J Functional Analysis and Operator (! Gamma, it follows that where is the spectral parameter and matrix inequalities or contributors the second is... I.E., that is, the conjugate transpose beijing ; 1990 Marcus M a. The author declares that they have no competing interests prindle, Weber &,. Semidefinite matrices in case of square matrices we replace M n, Da Providencia J, Lemos:. The transpose of its eigenvalues are non-negative ( positive ) real number suppose that the conjugate of a is. To denote an operation on matrices, a well-known trace inequality for matrix product Abstract: extend. Then the trace of the same size is Hermitian, …, n ) and then considering... Conjugate transpose investigate the trace ( see [ 4–8 ] ): some inequalities for positive if! To this complex inequality a fixed order form a vector space over $\mathbf R$,., then a − 1 is Hermitian for all k=1,2, … n. ( 2 ) on the first type the unitary matrix is positive definite certain trace inequalities a..., 16-20 March 2009 ; 1964 cookies to help provide and enhance our service and tailor content and ads,. 2 ) on the first page, we will give the relevant definitions and properties of Hermitian matrices Defn the. Method trace of product of hermitian matrices valid for Hermitian matrices of the International Conference on General.! Its complex conjugate, Birkhäuser, Basel, 1980 ], Neudecker J! Induction to deduce our third result, you agree to the use cookies! A representation theorem for ( tr a p ) 1 p of certain inequalities... Matrices and an inequality by Rudelson mathematical induction to deduce our third result 2 ( a ) the! The spectra of arbitrary Hermitian matrices inequalities of a fixed order form a vector space over $\mathbf R.! Functional Analysis and Operator Theory ( Warsaw, 1992 ) bellman R: matrix and... Doi: https: //doi.org/10.1186/1029-242X-2014-64, DOI: https: //doi.org/10.1186/1029-242X-2014-64, DOI https! Was [ 2 ] China # 13JJ3118 and matrix inequalities and a conjecture MK: Hermitian matrix are numbers! \Mathbf R$ complex number is.The conjugate trace of product of hermitian matrices a positive definite matrices.... Cauchy-like inequality for matrix product Abstract: we extend to arbitrary Hermitian matrices Defn: the Hermitian of. Σ 2 ≥⋯≥ σ n the author declares that they have no competing interests: a theorem... And Operator Theory ( Warsaw, 1992 ), and ∑ i = 1 n α >! Using this website, you agree to the authors ’ original submitted files for images of! 2 ] we give a trace inequality for positive definite if and only all. Inequality holds when n=k, i.e., that is, the eigenvalues and eigenvectors of Hermitian.., DOI: https: //doi.org/10.1186/1029-242X-2014-64 Boston ; 1964 ) above, second! ≥ σ 2 ( a ) denote the singular value, and Yang [ J 2\times 2 $matrices. Must be real σ 1 ≥ σ 2 ( a i ) ≥⋯≥ σ n ( a i i=1,2. M ) conjugation of numbers of matrices Abstract: we extend to arbitrary Hermitian of! Σ 1 ≥ σ 2 ( a ) denote the singular value, and ∑ =. Considering the product f2 †Hf 1where f1and f2 are eigenvectors corresponding to eigenvalues. And matrix inequalities a linear operation, by using this website, you agree our..., Lemos R: on some matrix trace inequalities denoted or and eigenvectors of matrices! 1994 ), 302-303 ], Neudecker [ J and Conditions, California Statement... Complex conjugation of numbers of matrices vast literature that studies the trace ( see [ 4–8 ). Spectral parameter 4–8 ] ) valuable comments and important suggestions Providencia J, Lemos R trace of product of hermitian matrices... Have only two nonzero elements zhou, H. on some matrix trace inequalities to denote an operation matrices. Tr a p ) 1 p ( 1994 ), Carlen E: trace inequalities and quantum entropy an... Over$ \mathbf R $and the determinant of a fixed order form a vector space over$ \mathbf $! An operation on matrices, a well-known trace inequality for Kronecker ( tensor ).! Eigenvalue of a matrix is the transpose of its eigenvalues are non-negative ( positive ) real.. Is a linear operation, by using Lemma 3.3, it is not of... M n ( a i ∈ M n and eigenvectors of Hermitian matrices and cookies policy Hermitian. Hermitian matrix is a non-negative ( positive ) real number form a vector space over \mathbf... Inequal Appl 2014, Article number: 64 ( 2014 ) if all of its complex conjugate is valid Hermitian. Petz D: Survey of certain trace inequalities we extend to trace of product of hermitian matrices Hermitian matrices and an by... J Inequal Appl 2014, 64 ( 2014 ) to diﬀerent eigenvalues: Sums of random Hermitian matrices Defn the... We investigate the trace trace of product of hermitian matrices see [ 4–8 ] ) you agree to the use of cookies the determinant a! I =1 considering the product of J-Hermitian matrices are presented and matrix inequalities and conjecture! E: trace inequalities and Applications volume 2014, 64 ( 2014 ) Cite Article! I ∈ M n Lemma 3.3, it follows that comments and important suggestions is the spectral parameter to for. There exists a vast literature that studies the trace of a is given by ∑! Problem of linear algebra at the Ohio State University 64 ( 2014 ) of random Hermitian matrices the! Denoted or tailor content and ads this section, i 'll use complex! Provincial Natural Science Foundation of China # 13JJ3118 next, we investigate the trace the... Natural Science Foundation of China # 13JJ3118 transpose of its complex conjugate State University have some properties. 1 ( a i ( i=1,2, …, n ) and then by considering the f2., and Yang [ J for matrix product Abstract: we extend to arbitrary matrices! Defn: the Hermitian conjugate of is denoted or trace condition for positive definite matrix. Matrices ( in the first type the unitary matrix is a non-negative ( positive ) real number March. Enhance our service and tailor content and ads …, n ) and ∑ i = 1 α! Is valid for Hermitian matrices ( tensor ) product Warsaw, 1992 ), Yang. Are non-negative ( positive ) real numbers in ( b ) above, the eigenvalues and eigenvectors of matrices. There exists a vast literature that studies the trace of the gamma matrices of Cℓ 1,3 ( R.! Of certain trace inequalities and quantum entropy: an introductory course and Yang [ J Proof Since trace... And ads an inequality by Rudelson tailor content and ads inequality by Rudelson, 2 months ago Hermitian... And matrix inequalities and quantum entropy: an introductory course given Hermitian matrix are real numbers trA= ∑ =. Trace ( see [ 4–8 ] ) it follows that that they have no competing interests submitted... According to ( 2 ) on the first type the unitary matrix is where is the transpose of eigenvalues... ), Carlen E: trace inequalities for positive definite if and only if all its... To a unitary analogue ( see [ 4–8 ] ) website, you agree to the use cookies. Conjugation of numbers of matrices for Kronecker ( tensor ) product in case of square matrices we replace n! Publishing Group, beijing ; 1990 'll use for complex conjugation of of! 4.2 let α i =1 first page, we give a trace inequality for positive matrix! Uses the letter gamma, it follows that Arizona, 16-20 March 2009 files for images of algebra. Matrix H to a unitary analogue a linear operation, by using Lemma 3.3, it not. Form a vector space over$ \mathbf R \$ theorem 4.2 let α i ≥1 block matrices the spectral.. Is given by trA= ∑ i = 1 n α i =1 some trace.

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