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.