Parametrizing spectral filters for nonsingular polynomial matrices

Abstract

Two polynomial matrices are spectrally equivalent if they have the same finite and infinite elementary divisors. Spectral equivalence can be characterized using very special polynomial matrices called spectral filters. The main results in this paper concern the parametrization of spectral filters for any given pair of nonsingular spectrally equivalent polynomial matrices of arbitrary degree d. First we parametrize the set of their spectral filters of degree less than d when the given spectrally equivalent matrices have nonsingular leading coefficients. The parameter space is the subset of invertible matrices of the centralizer of any of their linearizations. When the leading coefficients are singular, we provide a parametrization of the set of spectral filters of degree d−1 and show how to obtain the spectral filters of degree less than d−1. Now the parameter space is the subset of invertible matrices of the centralizer of any linearization of the reversals with respect to a scalar that is not an eigenvalue of the given matrices. Ideas from the Theory of Linear Control Systems are used to introduce left and right realizations of matrix polynomials. These are the building blocks on which the parameterization of spectral filters is constructed. The results are illustrated with examples.