Canonical matrices of bilinear and sesquilinear forms

TL;DR

This paper provides canonical matrices for various bilinear and sesquilinear forms over different fields and involutions, using a representation theory-based reduction method to classify systems of forms and mappings.

Contribution

It introduces a unified approach to derive canonical matrices for forms over multiple fields and involutions, extending previous classification methods.

Findings

Canonical matrices for forms over algebraically closed and real closed fields

Canonical matrices for sesquilinear forms over quaternions with involution

Reduction method based on representation theory for classifying systems of 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].