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

TL;DR

This paper demonstrates that positive definite matrices can be congruently transformed into diagonal forms using pseudo-orthogonal, pseudo-unitary, and symplectic matrices, extending the Schweinler-Wigner extremum principle to broader classes of bases.

Contribution

It provides a unified approach to congruence transformations for positive matrices and generalizes the Schweinler-Wigner orthogonalization method to pseudo-orthogonal and symplectic bases.

Findings

Matrices are congruent to diagonal forms via pseudo-orthogonal/unitary matrices.

Proof techniques can be adapted to Williamson's theorem for symplectic matrices.

Generalization of Schweinler-Wigner method for basis construction.

Abstract

It is shown that a $N\times N$ 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 $Sp(N,{\cal R})$ [$Sp(N,{\cal C})$]. 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.