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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]

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]