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].

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

References in corpus (5)

• Canonical forms for complex matrix congruence and *congruence
• 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
• Canonical matrices of isometric operators on indefinite inner product spaces

Cited by in corpus (11)

• Congruence of multilinear forms
• Congruence of matrix spaces, matrix tuples, and multilinear maps
• Matrices that are self-congruent only via matrices of determinant one
• Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
• 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
• Classification of linear mappings between indefinite inner product spaces
• Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
• Lipschitz property for systems of linear mappings and bilinear forms
• The geometric classification of $2$-step nilpotent algebras and applications
• Classification of sesquilinear forms with the first argument on a subspace or a factor space
• Canonical matrices of forms and pairs of forms over finite and p-adic fields