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The signed enhanced principal rank characteristic sequence (sepr-sequence) of an Hermitian matrix is the sequence , where is either , , , , , , or , based on the following criteria: if B has both a positive and a negative order-k principal minor, and each order-k principal minor is nonzero. (respectively, ) if each order-k principal minor is positive (respectively, negative). if each order-k principal minor is zero. if B has each a positive, a negative and a zero order-k principal minor. (respectively, ) if B has both a zero and a nonzero order-k principal minor, and each nonzero order-k principal minor is positive (respectively, negative). Such sequences provide more information than the epr-sequence in the literature, where the kth term is either , , or based on whether all, none, or some (but not all) of the order-k principal minors of the matrix are nonzero. Various sepr-sequences are shown to be unattainable by Hermitian matrices. In particular, by applying Muir’s law of extensible minors, it is shown that subsequences such as and are prohibited in the sepr-sequence of a Hermitian matrix. For Hermitian matrices of orders , all attainable sepr-sequences are classified. For real symmetric matrices, a complete characterization of the attainable sepr-sequences whose underlying epr-sequence contains as a non-terminal subsequence is established. [ABSTRACT FROM AUTHOR]

In this paper, the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equations and the Hermitian nonnegative definite solutions to the system of matrix equations are, respectively, put forward, by making full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are, respectively, presented. [ABSTRACT FROM PUBLISHER]

In this note, we characterize linear maps between spaces of Hermitian matrices over a field with involution which preserve the rank of the identity matrix and the rank of tensor products of rank one Hermitian matrices. It is assumed that the fixed field with respect to the field with involution has at least four elements. We also present some applications of our main result to other preserver problems. [ABSTRACT FROM AUTHOR]

Using the method of seeds and branch duplication, it is shown that for every tree of diameter, there is an Hermitian matrix with as few asdistinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter, it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of. Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made. [ABSTRACT FROM PUBLISHER]

We employ the implicit finite difference scheme with the shifted Grünwald formula to discretize the fractional diffusion equations with constant coefficients. The coefficient matrix possesses the positive definite Toeplitz-like structure, so we can use the Hermitian and skew-Hermitian splitting method for solving the system. Krylov subspace methods with circulant preconditioners such as Strang’s and T. Chan’s preconditioners are proposed to solve each subsystem via using the fast Fourier transforms (FFTs). Moreover, we present convergence analysis and prove the spectrum of the preconditioned matrices to be clustered around 1. Superlinear convergence rates of the proposed algorithms are obtained. Numerical results illustrate the effectiveness and robustness of circulant preconditioners. [ABSTRACT FROM AUTHOR]

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the 'qubit portrait' bound for the distance between the qutrit states are considered in explicit form. [ABSTRACT FROM AUTHOR]

In this paper, we present several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing interior-point methods (IPMs) for semidefinite optimization (SDO). [ABSTRACT FROM AUTHOR]

Abstract: We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms. [Copyright &y& Elsevier]

Abstract: We show some majorization inequalities and apply them to derive norm, eigenvalue, singular value, and trace inequalities of matrices. We also present a generalization and a different proof of a recent result on majorization concerning partitioned Hermitian positive semidefinite matrices. [Copyright &y& Elsevier]

A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2 × 2 and 3 × 3, as well as k × k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold. [ABSTRACT FROM PUBLISHER]

Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger's papers on the classical moment problem. In this paper, we obtain an analog of Hamburger's criterion for the matrix moment problem. [ABSTRACT FROM AUTHOR]

We derive some properties of complex Hessenberg matrices and use the relevant normal matrix cases to examine the lengths of Ritz vectors in the rounding error analysis of the Lanczos tridiagonalization process. This question is important for the computational use of the process and has already been studied for the real symmetric matrix case, but because of its intricate and unedifying nature, part of the theory was never submitted to scientific journals. We develop a new and more palatable theory which also applies to Lanczos processes adapted to any form of normal matrix with collinear eigenvalues such as a Hermitian or skew-Hermitian matrix. The nonnormal matrix properties are intended to help in the analysis of the unsymmetric Lanczos process. [ABSTRACT FROM AUTHOR]

Abstract: The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation , as well as the extremal inertias of a partial block Hermitian matrix. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain some new inequalities by means of the mean inequalities of random variables, which include generalizations of the Greub–Rheinboldt inequality. [Copyright &y& Elsevier]

We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems. [ABSTRACT FROM AUTHOR]

We further generalize the technique for constructing the Hermitian/skew-Hermitian splitting (HSS) iteration method for solving large sparse non-Hermitian positive definite system of linear equations to the normal/skew-Hermitian (NS) splitting obtaining a class of normal/skew-Hermitian splitting (NSS) iteration methods. Theoretical analyses show that the NSS method converges unconditionally to the exact solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the NSS iteration which is dependent solely on the spectrum of the normal splitting matrix, and is independent of the eigenvectors of the matrices involved. We present a successive-overrelaxation (SOR) acceleration scheme for the NSS iteration, which specifically results in an acceleration scheme for the HSS iteration. Convergence conditions for this SOR scheme are derived under the assumption that the eigenvalues of the corresponding block Jacobi iteration matrix lie in certain regions in the complex plane. A numerical example is used to show that the SOR technique can significantly accelerate the convergence rate of the NSS or the HSS iteration method. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We give a simple proof of a determinantal inequality due to Marcus (1976). [ABSTRACT FROM AUTHOR]