Spectral decomposition of $(\star ,{\epsilon }_1,{\epsilon }_2)$-structured matrix polynomials

We provide the spectral decompositions of $(\star ,{\epsilon }_1,{\epsilon }_2)$-structured matrix polynomials $P(\lambda )$ in the unified form by a standard pair $(X,J)$ and a parameter matrix $\mathrm{\Gamma }$. Using the recursive relationship between the coefficient matrices of $P(\lambda )$, equivalent expressions of these coefficient matrices are provided. When $J$ is assumed to be a block diagonal matrix, we show that the parameter matrix $\mathrm{\Gamma }$ has a special structure.