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

Abstract

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 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.

On classification of normal matrices in indefinite inner product spaces

“…hypothesis that the spectrum is real. For the case of a nonreal spectrum, further considerations are necessary, see [18] for details.…”

“…We mainly consider the case F = C here, but we will extend results to the real case, whenever this easily achievable. However, the investigation of the real case in full detail needs additional discussions and is referred to the subsequent paper [18].…”