Congruences and canonical forms for a positive matrix: Application to the Schweinler–Wigner extremum principle

Abstract

It is shown that a N×N real symmetric [complex Hermitian] positive definite matrix V is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in SO(m,n)[SU(m,n)], for any choice of partition N=m+n. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson’s theorem which states that if N is even then V is congruent also to a diagonal matrix modulo a symplectic matrix in Sp(N,R)[Sp(N,C)]. Applications of these results considered include a generalization of the Schweinler–Wigner method of “orthogonalization based on an extremum principle” to construct pseudo-orthogonal and symplectic bases from a given set of linearly independent vectors.