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

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 $\mathrm{\Gamma }$ 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.