Essential decomposition of polynomially normal matrices in real indefinite inner product spaces
Christian Mehl · Electronic Journal of Linear Algebra · 2006
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 allselfadjoint, skew-adjoint, and unitary matrices, but that is small enoughsuchthat all elementscan be completely classified. The essential decomposition of a real polynomially normal matrix isintroduced. This is a decomposition into three parts, one part having real spectrum only and twoparts that can be described by two complex matrices that are polynomially normal with respect toa sesquilinear and bilinear form, respectively. In the paper, the essential decomposition is used as atool in order to derive a sufficient condition for existence of invariant semidefinite subspaces and toobtain canonical forms for real polynomially normal matrices. In particular, canonical forms for realmatrices that are selfadjoint, skewadjoint, or unitary with respect to an indefinite inner product arerecovered.