Spectral decomposition of $(\star, \epsilon)$-palindromic matrix polynomial and its applications

TL;DR

This paper develops a spectral decomposition method for $(igstar,igepsilon)$-palindromic quadratic matrix polynomials using standard pairs and parameter matrices, enabling solutions to inverse eigenvalue and eigenvalue embedding problems.

Contribution

It introduces a spectral decomposition approach for $(igstar,igepsilon)$-palindromic matrix polynomials and applies it to solve inverse eigenvalue and eigenvalue embedding problems.

Findings

Spectral decomposition achieved via standard pairs and parameter matrices.

Special structure of the parameter matrix when $J$ is block diagonal.

Application to inverse eigenvalue and eigenvalue embedding problems with no spill-over.

Abstract

This paper provides the spectral decomposition of $(\star,\epsilon)$-palindromic quadratic matrix polynomial $P(\lambda)$ by a standard pair and a parameter matrix. When $J$ is assumed to be a block diagonal matrix, the parameter matrix $\Gamma$ has a special structure. And then the spectral decomposition is applied to solve the inverse eigenvalue problem and the eigenvalue embedding problem with no spill-over.