Spectral decomposition of selfadjoint matrices in positive semidefinite inner product spaces and its applications

Abstract

Given a positive semidefinite matrix A, we obtain spectral decomposition of A-selfadjoint matrices which act on a positive semidefinite inner product space in which the inner product is induced by A. As applications, we improve the singular value decomposition of square matrices given by Zheng and Li (2014) and give polar decomposition of square matrices in positive semidefinite inner product spaces, and these two decompositions are extended to rectangular matrices.