Generalized Fiedler Pencils for Rational Matrix Functions
Rafikul Alam, Namita Behera · SIAM Journal on Matrix Analysis and Applications · 2018
We introduce generalized Fiedler (GF) pencils for an $n\times n$ rational matrix function $G(\lambda)$ for computing eigenvalues and poles of $G(\lambda)$ and show that the GF pencils are linearizations of $G(\lambda).$ These GF pencils generalize the GF pencils of matrix polynomials to the case of rational matrix functions. We show that the GF pencils of $G(\lambda)$ provide a class of linearizations of $G(\lambda)$ which contains potential candidates for structure-preserving linearizations of $G(\lambda)$. Indeed, we describe construction of a self-adjoint GF pencil of $G(\lambda)$ when $G(\lambda)$ is self-adjoint. We further show that the GF pencils of $G(\lambda)$ allow operation-free recovery of eigenvectors of $G(\lambda),$ that is, the eigenvectors of $G(\lambda)$ can be recovered from those of the GF pencils without performing any arithmetic operations. We also introduce GF pencils for the Rosenbrock system polynomial $\mathcal{S}(\lambda)$ associated with a linear time-invariant (LTI) state-space system and show that the GF pencils are trimmed structure-preserving linearizations of $\mathcal{S}(\lambda).$ We analyze the recovery of eigenvectors of $\mathcal{S}(\lambda),$ which are known as the invariant zero directions of the LTI system, from the GF pencils of $\mathcal{S}(\lambda)$.