Structured Linearizations of Structured Rational Matrices

TL;DR

This paper introduces a family of structured linearizations called GFPR that preserve spectral and structural properties of rational matrices, enabling more stable and accurate numerical computations.

Contribution

It develops the GFPR framework for structured linearizations of rational matrices, ensuring preservation of symmetry, skew-symmetry, and other properties, which was not previously available.

Findings

GFPR are valid linearizations for rational matrices.

GFPR can generate structured linearizations for symmetric, skew-symmetric, T-even, and T-odd matrices.

Structured linearizations improve numerical stability and physical relevance.

Abstract

Numerical computations involving rational matrices often benefit from preserving underlying matrix structures such as symmetry, Hermitian properties, or sparsity that reflect physical, geometric, or algebraic characteristics of the system. Maintaining such structures enhances stability, accuracy, and efficiency. Linearization, a technique that reformulates rational matrix problems as generalized eigenvalue problems (GEPs) of larger matrices, is widely used but does not automatically retain structure. In this chapter, we focus on structured linearizations, which preserve both the spectral information of the original rational matrix and its intrinsic structural properties. To achieve this, we present the construction of a family of linearizations called generalized Fiedler pencils with repetition (GFPR), which we prove to be valid linearizations for rational matrices. Moreover, we demonstrate that the GFPR family serves as a versatile framework for generating structured linearizations, specifically symmetric, skew-symmetric, T-even, and T-odd linearizations, provided the original rational matrix exhibits the corresponding structure. These structured linearizations facilitate the use of specialized, structure-preserving algorithms, reduce numerical errors, and yield physically meaningful solutions in application