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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]

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]