Classification problems for system of forms and linear mappings

Abstract

We devise a method that reduces the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings. Canonical matrices of (i) bilinear or sesquilinear forms, (ii) pairs of symmetric, skew-symmetric, or Hermitian forms, (iii) isometric or selfadjoint operators on a space with nonsingular symmetric, or skew-symmetric, or Hermitian form are obtained over any field of characteristic not 2 up to classification of Hermitian forms over its finite extensions.

46 pages

Cited by in corpus (15)

• Canonical matrices for linear matrix problems
• Canonical forms for complex matrix congruence and *congruence
• Complexity of matrix problems
• Congruences of a square matrix and its transpose
• Canonical matrices of bilinear and sesquilinear forms
• A regularization algorithm for matrices of bilinear and sesquilinear forms
• Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild
• Canonical matrices of isometric operators on indefinite inner product spaces
• The problems of classifying pairs of forms and local algebras with zero cube radical are wild
• Pairs of mutually annihilating operators
• Congruence of multilinear forms
• 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 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