Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle

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.