On classification of normal matrices in indefinite inner product spaces

Abstract

Canonical forms are developed for several sets of matrices that are normal withrespect 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 originalmatrix. From this result, canonical forms for complex matrices that are selfadjoint, skewadjoint, orunitarywith respect to the given indefinite inner product are derived. Most of the canonical formsfor the latter three special types of normal matrices are known in the literature, but it is the aimof this paper to present a general theorythat allows the unified treatment of all different cases andto collect known results and new results such that all canonical forms for the complex case can befound in a single source.