Canonical matrices of bilinear and sesquilinear forms

Abstract

Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (c) sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F. A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].

arXiv comment44 pages; misprints corrected; accepted for publication in Linear Algebra and its Applications (2007)

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Cited by in corpus (22)

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• Miniversal deformations of matrices of bilinear forms
• 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
• Wildness for tensors
• Representations of quivers and mixed graphs
• On quantum groups associated to non-Noetherian regular algebras of dimension 2
• Congruence of multilinear forms
• Congruence of matrix spaces, matrix tuples, and multilinear maps
• Classification of linear operators satisfying $(Au,v)=(u,A^{r}v)$ or $(Au,A^{r}v)=(u,v)$ on a vector space with indefinite scalar product
• Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
• Hochschild products and global non-abelian cohomology for algebras. Applications
• Matrices that are self-congruent only via matrices of determinant one
• Classification of linear mappings between indefinite inner product spaces
• Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
• Metabelian associative algebras
• Lipschitz property for systems of linear mappings and bilinear forms
• Classification of sesquilinear forms with the first argument on a subspace or a factor space
• The geometric classification of $2$-step nilpotent algebras and applications
• Topological classification of sesquilinear forms: reduction to the nonsingular case
• On a type of commutative algebras
• Canonical matrices of forms and pairs of forms over finite and p-adic fields