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

Cited by in corpus (29)

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• Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
• Miniversal deformations of matrices under *congruence and reducing transformations
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• Canonical matrices of isometric operators on indefinite inner product spaces
• Two-generated algebras and standard-form congruence
• Representations of quivers and mixed graphs
• Tensors with maximal symmetries
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• Congruence of multilinear forms
• Congruence of matrix spaces, matrix tuples, and multilinear maps
• Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms
• Matrices that are self-congruent only via matrices of determinant one
• Classification of linear mappings between indefinite inner product spaces
• Finsler geometries on strictly accretive matrices
• Classification of sesquilinear forms with the first argument on a subspace or a factor space
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• Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras
• Topological classification of sesquilinear forms: reduction to the nonsingular case
• Modeling complex points up to isotopy
• More on regular subgroups of the affine group
• Projective Self-dual polygons in higher dimensions
• On rank one 2-representations of web categories
• Topological classification of systems of bilinear and sesquilinear forms
• Canonical matrices of forms and pairs of forms over finite and p-adic fields
• CR singularities of real fourfolds in ${\mathbb{C}}^3$