On Some Inverse Eigenvalue Problems of Quadratic Palindromic Systems

TL;DR

This paper investigates inverse eigenvalue problems for quadratic palindromic systems, providing conditions for solvability and algorithms for solutions, with applications to entire or partial eigenpair data and model updating.

Contribution

It introduces new methods and conditions for solving inverse eigenvalue problems in quadratic palindromic systems, including algorithms and numerical illustrations.

Findings

Derived solvability conditions for inverse eigenvalue problems.

Proposed algorithms for solving inverse problems with numerical examples.

Addressed model updating problems with no-spillover.

Abstract

This paper concerns some inverse eigenvalue problems of the quadratic $\star$-(anti)-palindromic system $Q(\lambda)=\lambda^2 A_1^{\star}+\lambda A_0 + \epsilon A_1$, where $\epsilon=\pm 1$, $A_1, A_0 \in \mathbb{C}^{n\times n}$, $A_0^{\star}=\epsilon A_0$, $A_1$ is nonsingular, and the symbol $\star$ is used as an abbreviation for transpose for real matrices and either transpose or conjugate transpose for complex matrices. By using the spectral decomposition of the quadratic $\star$-(anti)-palindromic system, the inverse eigenvalue problems with entire/partial eigenpairs given, and the model updating problems with no-spillover are considered. Some conditions on the solvabilities of these problems are given, and algorithms are proposed to find these solutions. These algorithms are illustrated by some numerical examples.