On classification of normal matrices in indefinite inner product spaces

Abstract

Canonical forms are developed for several sets of matrices that are normal with
respect to an indefinite inner product induced bya nonsingular Hermitian, symmetric, or skewsymmetric matrix. The most general result covers the case of polynomially normal matrices, i.e.,
matrices whose adjoint with respect to the indefinite inner product is a polynomial of the original
matrix. From this result, canonical forms for complex matrices that are selfadjoint, skewadjoint, or
unitarywith respect to the given indefinite inner product are derived. Most of the canonical forms
for the latter three special types of normal matrices are known in the literature, but it is the aim
of this paper to present a general theorythat allows the unified treatment of all different cases and
to collect known results and new results such that all canonical forms for the complex case can be
found in a single source.