Transformations and inertia of solutions to linear matrix equations

Abstract

Linear equations and operators in a space of matrices are investigated. The transformations of matrix equations which allow one to find the conditions of solvability and the inertial properties of Hermite solutions are determined. New families of matrices (collectives) are used in the theory of inertia and positive invertibility of linear operators and, in particular, in the problems of localization of matrix spectra and matrix beams.