Unified framework for Fiedler-like strong linearizations of polynomial and rational matrices

Linearization is a widely used method for solving polynomial eigenvalue problems (PEPs) and rational eigenvalue problem (REPs) in which the PEP/REP is transformed to a generalized eigenproblem and then solve this generalized eigenproblem with algorithms available in the literature. Fiedler-like pencils (Fiedler pencils (FPs), generalized Fiedler pencils (GFPs), Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs)) are well known classes of strong linearizations. GFPs are an intriguing family of linearizations, and GF pencils are the fundamental building blocks of FPRs and GFPRs. As a result, FPRs and GFPRs have distinctive features and they provide structure-preserving linearizations for structured matrix polynomials. But GFPRs do not use the full potential of GF pencils. Indeed, not all the GFPs are FPRs or GFPRs, and vice versa. The main aim of this paper is two-fold. First, to build a unified framework for all the Fiedler-like pencils FPs, GFPs, FPRs and GFPRs. To that end, we construct a new family of strong linearizations (named as EGFPs) of a matrix polynomial $P(\lam)$ that subsumes all the Fiedler-like linearizations. A salient feature of the EGFPs family is that it allows the construction of structured preserving banded linearizations with low bandwidth for structured (symmetric, Hermitian, palindromic) matrix polynomial. Low bandwidth structured linearizations may be useful for numerical computations. Second, to utilize EGFPs directly to form a family of Rosenbrock strong linearizations of an $n \times n$ rational matrix $G(\lam)$ associated with a realization. We describe the formulas for the construction of low bandwidth linearizations for $P(\lam)$ and $G(\lam)$. We show that the eigenvectors, minimal bases/indices of $P(\lam)$ and $G(\lam)$ can be easily recovered from those of the linearizations of $P(\lam)$ and $G(\lam)$.