Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form

Abstract

Let V be a vector space over a field F with scalar product given by a nondegenerate sesquilinear form whose matrix is diagonal in some basis. If F=C, then we give canonical matrices of isometric and selfadjoint operators on V using known classifications of isometric and selfadjoint operators on a complex vector space with nondegenerate Hermitian form. If F is a field of characteristic different from 2, then we give canonical matrices of isometric, selfadjoint, and skewadjoint operators on V up to classification of symmetric and Hermitian forms over finite extensions of F.