The Congruent Centralizer of a Block Diagonal Matrix

Abstract

Let a complex matrix A be the direct sum of its square submatrices B and C that have no common eigenvalues. Then every matrix X belonging to the centralizer of A has the same block diagonal form as the matrix A. This paper discusses how the conditions on the submatrices B and C should be modified for an analogous assertion about the congruent centralizer of A, which is the set of matrices X such that X^*AX = A, to be valid. Also the question whether the matrices in the congruent centralizer are block diagonal if A is a block anti-diagonal matrix is considered. Bibliography: 2 titles.