Canonical forms for complex matrix congruence and *congruence

TL;DR

This paper provides a direct proof and explicit algorithms for canonical forms of complex matrices under congruence and *congruence, extending to pairs of matrices with applications in symmetric, skew-symmetric, and Hermitian cases.

Contribution

It offers a direct proof of known canonical forms and introduces explicit algorithms for their computation, with applications to pairs of matrices.

Findings

Explicit algorithms for canonical forms are provided.

Canonical pairs for simultaneous congruence are derived.

Applications include forms for pairs of symmetric, skew-symmetric, and Hermitian matrices.

Abstract

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices.