ELA, Volume 15, pp. 50-83, February 2006, abstract.
On Classification of Normal Matrices in Indefinite
Inner Product Spaces
Christian Mehl
Canonical forms are developed for several sets of matrices that
are normal with respect to an indefinite inner product induced
by a nonsingular Hermitian, symmetric, or skew-symmetric 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 unitary with 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 theory that 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.