Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

TL;DR

This paper introduces a unified method for solving the structure-preserving eigenvalue embedding problem in quadratic matrix polynomials with symmetry, enabling targeted perturbations while maintaining key invariance properties.

Contribution

It develops a comprehensive approach for structured eigenvalue embedding in quadratic polynomials, including no spillover perturbations, with analytical expressions applicable to real-world problems.

Findings

Analytical formulas for structure-preserving perturbations

Applicable to symmetric, Hermitian, and $ ext{star}$-even/odd polynomials

Numerical examples validate the theoretical results

Abstract

In this paper, we propose a unified approach for solving structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, $\star$-even and $\star$-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases. The obtained results are supported with numerical examples.