A “shift-and-deflate” technique for quadratic matrix polynomials

A technique for deflating two eigenvalues $\lambda_1$, $\lambda_2$ of an $n\times n$ quadratic matrix polynomial $A(z)$ is proposed. The method requires the knowledge of a right eigenvector corresponding to $\lambda_1$, and of a left eigenvector corresponding to $\lambda_2$. This technique consists of two stages: the shift of $\lambda_1$ and $\lambda_2$ to $0$ and $\infty$, respectively, and the deflation of $0$ and $\infty$. The final result is the construction of an $(n-1)\times (n-1)$ quadratic matrix polynomial, which shares with $A(z)$ all the eigenvalues, except for $\lambda_1$ and $\lambda_2$. The particular case of $*$-palindromic quadratic matrix polynomials is treated.