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 (37)

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• Complexity of matrix problems
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• A regularization algorithm for matrices of bilinear and sesquilinear forms
• 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
• 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
• Wildness for tensors
• The problems of classifying pairs of forms and local algebras with zero cube radical are wild
• Pairs of mutually annihilating operators
• Transformations of Stabilizer States in Quantum Networks
• Representations of quivers and mixed graphs
• A canonical form for nonderogatory matrices under unitary similarity
• Remarks on the classification of a pair of commuting semilinear operators
• Congruence of multilinear forms
• Congruence of matrix spaces, matrix tuples, and multilinear maps
• When does a linear map belong to at least one orthogonal or symplectic group?
• Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms
• 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
• 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
• Lipschitz property for systems of linear mappings and bilinear forms
• On nilpotent Chernikov $p$-groups with elementary tops
• Operators on positive semidefinite inner product spaces
• Classification of sesquilinear forms with the first argument on a subspace or a factor space
• Reduction of a pair of skew-symmetric matrices to its canonical form under congruence
• Topological classification of sesquilinear forms: reduction to the nonsingular case
• Decomposition of (co)isotropic relations
• Calabi-Yau algebras viewed as deformations of Poisson algebras
• Skew-adjoint maps and quadratic Lie algebras
• Canonical matrices of forms and pairs of forms over finite and p-adic fields