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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]

Without 'positive definiteness' demanded in the present papers, the forward and reverse inequalities for Hadamard products of any number of invertible Hermitian matrices are obtained, and the sufficient and necessary conditions for the equations in these inequalities are given. As Hermitian positive matrices naturally satisfy the added constraints, these results generalize and improve the corresponding results in the present papers. Beyond that, with no demand of 'positive definiteness', these forward and backward inequalities are not determined mutually any longer. [ABSTRACT FROM AUTHOR]

We obtain a new criterion for the complete indeterminacy of the classical matrix problem and of the Nevanlinna-Pick matrix problem in terms of the convergence of two matrix series. The elements of these series are positive matrices and are expressed by analogs of the classical Schur parameters. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss the minimal rank of when X is Hermitian by applying singular value decomposition and some rank equalities of matrices, and obtain a representation of the minimal rank. Based on the representation, we obtain necessary and sufficient conditions for the matrix equation to have Hermitian solutions. [Copyright &y& Elsevier]

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 investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank. [Copyright &y& Elsevier]

Abstract: When an edge is removed from an undirected graph, there is a limited change that can occur in the multiplicity of an eigenvalue of a Hermitian matrix with that graph. Primarily for trees, we identify the changes that can occur and characterize the circumstances under which they occur. This extends known results for the removal of vertices. A catalog of examples is given to illustrate the possibilities that can occur and to contrast the case of trees with that of general graphs. [ABSTRACT FROM AUTHOR]

Abstract: The extremal ranks of matrix expression of A − BXC with respect to X H = X have been discussed by applying the quotient singular value decomposition Q-SVD and some rank equalities of matrices in this paper. [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]

Abstract: It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form. [Copyright &y& Elsevier]

Suppose [image omitted] is a field with proper involution and of arbitrary characteristic. Additive maps, which do not increase rank-one on hermitian matrices with entries from [image omitted] are classified. The result is then used to classify additive maps that do not increase minimal rank on the set of symmetric elements, relative to general involution on a matrix algebra. [ABSTRACT FROM AUTHOR]

Abstract: In an earlier paper Srivastava [2005. Combinatorial Hermitian Algebras Derived from Latin Squares. J. Statist. Plann. Inference 129, 305–316], subsequently referred to as [S05], the author had presented an infinite class of algebras of Hermitian matrices. Each of these algebras were derived from Latin Squares. In this paper, we show how to obtain a unitary matrix (over the complex field) such that is a diagonal matrix, for all matrices belonging to a particular algebra. The notation is defined in the next paragraph. The said diagonal matrix is a matrix that contains the characteristic roots of . The matrix may be different for different algebras. Many other related results are also presented. The algebras are further discussed. How the Hermitian matrices arise in Quantum Mechanics (where every observable attribute of every physical phenomenon is in a sense characterized by a Hermitian matrix) is briefly described. The significance of the existence of Hermitian algebras and the application to Physics and Chemistry is pointed out. [Copyright &y& Elsevier]

Suppose A-BXB*, A-BX -X*B*, and A-BX +X*B* are three linear matrix expressions over the field of complex numbers, where A is an Hermitian or skew-Hermitian matrix. In this paper, we consider how to choose an Hermitian or skew-Hermitian matrix X such that A-BXB* have the maximal and minimal possible ranks, and how to choose X such that A - BX ± X*B* attain the minimal possible ranks. Some applications to Hermitian or skew-Hermitian solutions of matrix equations with symmetric patterns are also given. [ABSTRACT FROM AUTHOR]

Let A =( a ij ) be an n -by- n matrix. For any real µ, define the polynomial where l (s) is the number of inversions of the permutation s in the symmetric group Sn . We prove that P µ (A) is a strictly increasing function of µ ? [-1,1], for a Hermitian positive definite nondiagonal matrix A , whose graph is a tree. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an infinite class of algebras of Hermitian matrices of size , where n is a positive integer and where is a prime number or a power of a prime number (so that the Galois Field GF exists). For each such n, we obtain separate Hermitian algebras, no two of which mutually commute. [Copyright &y& Elsevier]

Given an n-by-n Hermitian matrix A and a real number λ, index i is said to be Parter (resp., neutral, downer) if the multiplicity of λ as an eigenvalue of the principal submatrix A(i) is one more (resp., the same, one less) than that in A. In case the multiplicity of λ in A is at least 2 and the graph of A is a tree, there are always Parter vertices. Our purpose here is to advance the classification of vertices and, in particular, to relate classification to the combinatorial structure of eigenspaces. Some general results are given and then used to deduce some rather specific facts not otherwise easily observed. Examples are given. [ABSTRACT FROM AUTHOR]

The article discusses the measurement of the orthogonal matrix polynomials with periodic coefficients of recurrence relations. It uses the diagonalization of the matrix transform to determine the matrices. It presents mathematical theorems that explain the problem of reconstructing the measure of a periodic sequence of coefficients.