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

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