Canonical Forms for Certain Matrices Under Unitary Congruence

Abstract

If A is a matrix with complex elements and if A = A_T (where A_T denotes the transpose of A), there exists a non-singular matrix P such that PAP_T = D is a diagonal matrix (see (3), for example). It is also true (see the principal result of (5)) that for such an A there exists a unitary matrix U such that UAU_T = D is a real diagonal matrix with nonnegative elements which is a canonical form for A relative to the given U, U_T transformation.