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Structured Linearizations of Structured Rational Matrices

Avisek Bist, Namita Behera

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

Structured Linearizations of Structured Rational Matrices

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
Paper Structure (10 sections, 13 theorems, 51 equations)

This paper contains 10 sections, 13 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Generalized Fiedler pencils with repetition
  3. Fiedler matrices
  4. Index tuples
  5. GFPRs are linearizations
  6. Structured Lineraizations
  7. Symmetric GFPRs
  8. T-even linearizations
  9. T-odd linearizations
  10. Skew-symmetric linearizations

Key Result

Lemma 2.13

Let $\mathbold{\alpha}$ be a permutation of $\{0,1,\ldots,m-1\}$ and let $L_{A}(\lambda)=\lambda M^{A}_{-m}-M^{A}_{\mathbold{\alpha}}$ be the Fiedler pencil of $A(\lambda)$ corresponding to $\mathbold{\alpha}$ Then, $L_{A}(\lambda)$ is a linearization of $A(\lambda)$. Hence, there exist unimodular m Further,

Theorems & Definitions (44)

  • Definition 2.1: Permutations & Sub-permutations, AB2018
  • Definition 2.2: Index Tuple, AB2018
  • Definition 2.3: BT2014VA2011
  • Example 2.4
  • Definition 2.5: BDFR2015
  • Example 2.6
  • Definition 2.7: Consecutions & inversions, DA2019Automatic
  • Example 2.8
  • Definition 2.9: BDFR2015, Matrix Assignment
  • Definition 2.10: GFPR of $G(\lambda)$
  • ...and 34 more