Mathematics – Rings and Algebras
Scientific paper
2010-08-26
Linear Algebra Appl. 436-5 (2012), 1385-1405
Mathematics
Rings and Algebras
32 pages (v4)
Scientific paper
10.1016/j.laa.2011.08.035
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.
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