Linearizations for Rational Matrix Functions and Rosenbrock System Polynomials

Abstract

Our aim in this paper is twofold: First, for solving rational eigenproblems we introduce linearizations of rational matrix functions and propose a framework for their constructions via minimal realizations. We also introduce Fiedler-like pencils for rational matrix functions and show that the Fiedler-like pencils are linearizations of the rational matrix functions. Second, for computing zeros of a linear time-invariant system $\Sigma$ in state-space form, we introduce a trimmed structured linearization, which we refer to as a Rosenbrock linearization, of the Rosenbrock system polynomial $\mathcal{S}(\lambda)$ associated with $\Sigma.$ Also we introduce Fiedler-like pencils for $\mathcal{S}(\lambda),$ describe an algorithm for their constructions, and show that the Fiedler-like pencils are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lambda).$