Canonical forms for complex matrix congruence and *congruence

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.

31 pages

References in corpus (3)

• Classification problems for system of forms and linear mappings
• Congruences of a square matrix and its transpose
• A regularization algorithm for matrices of bilinear and sesquilinear forms

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• Classification of sesquilinear forms with the first argument on a subspace or a factor space
• CR singularities of real fourfolds in ${\mathbb{C}}^3$