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

TL;DR

This paper extends the eigenvalue exclusion theorem to singular matrix polynomials within the $\\mathbb{DL}(P)$ vector space, showing that relevant spectral information can still be recovered from associated pencils.

Contribution

It generalizes the eigenvalue exclusion theorem to singular matrix polynomials and demonstrates that key spectral data can be obtained from pencils in $\mathbb{DL}(P)$ under certain conditions.

Findings

Eigenvalue exclusion theorem extended to singular polynomials.

Spectral data recoverable from pencils satisfying exclusion hypothesis.

Representation via Bezoutians is crucial for the proof.

Abstract

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].