Spectral decomposition of $(\star,ε_1,ε_2)$-structured matrix polynomials with arbitrary degree and its applications

Abstract

This paper provides the spectral decompositions of $(\star,ε_1,ε_2)$-structured matrix polynomials $P(λ)$ in the unified form by a standard pair and parameter matrix. Using the recursive relationship between the coefficient matrices of $P(λ)$, equivalent expressions of these coefficient matrices are provided. And then the spectral decomposition is applied to solve the inverse eigenvalue problem and the eigenvalue embedding problem with no spill-over.