When does a linear map belong to at least one orthogonal or symplectic group?

TL;DR

This paper characterizes when a linear map over a field belongs to an orthogonal or symplectic group, especially focusing on the challenging case of characteristic 2 fields, providing necessary and sufficient conditions.

Contribution

It extends known results by providing necessary and sufficient conditions for the existence of orthogonal or symplectic forms in characteristic 2 fields, including finite fields.

Findings

Conditions for existence of orthogonal/symplectic forms in char(K)<>2

Necessary and sufficient conditions in char(K)=2

Characterization of hyperbolic and non-hyperbolic quadratic forms in finite fields

Abstract

Given an endomorphism u of a finite-dimensional vector space over an arbitrary field K, we give necessary and sufficient conditions for the existence of a regular quadratic form (resp. a symplectic form) for which u is orthogonal (resp. symplectic). A solution to this problem being already known in the case char(K)<>2, our main contribution lies in the case char(K)=2. When char(K)=2, we also give necessary and sufficient conditions for the existence of a regular symmetric bilinear form for which u is orthogonal. In the case K is finite with characteristic 2, we give necessary and sufficient conditions for the existence of an hyperbolic quadratic form (resp. a regular non-hyperbolic quadratic form, resp. a regular non-alternate symmetric bilinear form) for which u is orthogonal.