The $\mathbb{DL}(P)$ vector space of pencils for singular matrix polynomials

Given a possibly singular matrix polynomial $P(z)$, we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space $\mathbb{DL}(P)$ introduced in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl. 28(4), 971-1004, 2006] are related to those of $P(z)$. If $P(z)$ is regular, it is known that those pencils in $\mathbb{DL}(P)$ satisfying the generic assumptions in the so-called eigenvalue exclusion theorem are strong linearizations for $P(z)$. This property and the block-symmetric structure of the pencils in $\mathbb{DL}(P)$ have made these linearizations among the most influential for the theoretical and numerical treatment of structured regular matrix polynomials. However, it is also known that, if $P(z)$ is singular, then none of the pencils in $\mathbb{DL}(P)$ is a linearization for $P(z)$. In this paper, we prove that despite this fact a generalization of the eigenvalue exclusion theorem holds for any singular matrix polynomial $P(z)$ and that such a generalization allows us to recover all the relevant quantities of $P(z)$ from any pencil in $\mathbb{DL}(P)$ satisfying the eigenvalue exclusion hypothesis. Our proof of this general theorem relies heavily in the representation of the pencils in $\mathbb{DL} (P)$ via B\'{e}zoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181-209, 2015].