Classification problems for system of forms and linear mappings

TL;DR

This paper introduces a method to simplify the classification of systems of forms and linear mappings by reducing it to the classification of linear mappings, providing canonical forms over fields with characteristic not 2.

Contribution

It presents a unified approach to classify various forms and operators by linking them to linear mappings and canonical matrices, extending classification to fields with characteristic not 2.

Findings

Canonical matrices for bilinear and sesquilinear forms derived

Classification of pairs of symmetric, skew-symmetric, or Hermitian forms achieved

Canonical forms for operators on spaces with specific forms established

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.