Classification Problems for Systems of Forms and Linear Mappings

Abstract

A method is proposed that allow the reduction of many classification problems of linear algebra to the problem of classifying Hermitian forms. Over the complex, real, and rational numbers classifications are obtained for bilinear forms, pairs of quadratic forms, isometric operators, and selfadjoint operators.Many problems of linear algebra can be formulated as problems of classifying the representations of a quiver. A quiver is, by definition, a directed graph. A representation of the quiver is given (see [6], and also [2,14]) by assigning to each vertex a vector space and to each arrow a linear mapping of the corresponding vector spaces. For example, the quivers 1 e e 1 * * 4 4 2 1 9 9 e e correspond respectively to the problems of classifying:• linear operators (whose solution is the Jordan or Frobenius normal form),• pairs of linear mappings from one space to another (the matrix pencil problem, solved by Kronecker), and

Existence of an invariant form under a linear map

“…Suppose that the linear map T admits an invariant non-degenerate hermitian, resp skew-hermitian, form H. Then the necessary condition follows from existing literatures, for example see [Wal63,Ser87,Ser08].…”

On nilpotent Chernikov p-groups with elementary tops

“…Combining this result with those from [8], we get a complete classification of pairs of skew-symmetric bilinear forms. We denote by A the set of all pairs R + , where R ∈ {R f , R ∞,d , R −,d }, and by F the set of functions κ : A → Z ≥0 such that κ(A) = 0 for almost all A.…”

“…Recall this relation [8]. A representation R of K n over a field k consists of two finite dimensional vector spaces R(1) and R(2) and n linear maps…”

“…One can check (cf. [8]) that a representation R is isomorphic to a self-dual one if and only if there is a self-dual isomorphism f : R → R * , i.e. such an isomorphism that f (2) = f (1) * .…”

Essential decomposition of polynomially normal matrices in real indefinite inner product spaces

“…Remark 5.6. A result in the direction of Theorem 5.5 for the case δ = −1 has been obtained in [21] concerning existence of a decomposition of H-unitary matrices into indecomposable blocks. Also, the possible Jordan canonical forms of the indecomposable blocks have been fully described.…”