Symmetric matrices with respect to sesquilinear forms Original Research Article

Simple forms are obtained for matrices that are symmetric with respect to degenerate sesquilinear forms on finite dimensional complex linear spaces of column vectors. Symmetric matrices and the sesquilinear forms are then representable as block diagonals having simple forms as the diagonal blocks. The notion of indecomposability for symmetric matrices is studied. An example shows that, in contrast with the nondegenerate sesquilinear forms, an indecomposable symmetric matrix with respect to a degenerate sesquilinear form may have arbitrarily many Jordan blocks. All indecomposable symmetric matrices are characterized in two situations: when the sesquilinear form has only one degree of degeneracy, and when the form is semidefinite.