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 $\Gamma$. 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 $\Gamma$ has a special structure.
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