Essential decomposition of polynomially normal matrices in real indefinite inner product spaces

Main Article Content

Christian Mehl

Abstract

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 enoughsuchthat 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.

Article Details

Section
Article