ELA, Volume 15, pp. 84-106, February 2006, abstract.

Essential Decomposition of Polynomially Normal Matrices 
in Real Indefinite Inner Product Spaces

Christian Mehl

Polynomially normal matrices in real indefinite inner product 
spaces are studied, i.e., matrices whose adjoint with respect 
to the indefinite inner product is a polynomial in the matrix. 
The set of these matrices is a subset of indefinite inner product 
normal matrices that contains all selfadjoint, skew-adjoint, 
and unitary matrices, but that is small enough such that all 
elements can be completely classified. The essential decomposition 
of a real polynomially normal matrix is introduced. This is a 
decomposition into three parts, one part having real spectrum 
only and two parts that can be described by two complex matrices 
that are polynomially normal with respect to a sesquilinear and 
bilinear form, respectively. In the paper, the essential decomposition 
is used as a tool in order to derive a sufficient condition for 
existence of invariant semidefinite subspaces and to obtain canonical 
forms for real polynomially normal matrices. In particular, canonical 
forms for real matrices that are selfadjoint, skewadjoint, or unitary
with respect to an indefinite inner product are recovered.