论文标题
$ \ mathbb {dl}（p）$ pencils singular矩阵多项式的矢量空间
The $\mathbb{DL}(P)$ vector space of pencils for singular matrix polynomials
论文作者
论文摘要
如果可能是一个可能单数的矩阵多项式$ p（z）$，我们研究了矢量空间中的特征值，特征向量，根系多项式，最小指数和最小铅笔基础，$ \ mathbbbb {dl}（dl}（p）$中引入了mackey，mackey，mackey，mackey，mackey，mehl and mehl and mehl and mehl，shrmann。应用。 28（4），971-1004，2006]与$ p（z）$相关。如果$ p（z）$是常规的，则众所周知，这些铅笔$ \ m athbb {dl}（p）$满足所谓的特征值排除定理中的通用假设是$ p（z）$的强线性化定理。在$ \ mathbb {dl}（p）$中，铅笔的这种特性和块对称结构使这些线性化是对结构化常规矩阵多项式的理论和数值处理最大的影响。但是，还知道，如果$ p（z）$是单数的，那么$ \ mathbb {dl}（p）$中的铅笔都不是$ p（z）$的线性化。在本文中，我们证明，尽管这一事实的概括性排除定理对任何单个矩阵多项式$ p（z）$都具有概括性，并且这样的概括使我们能够从$ \ \ mathbb {dl}（p）（p）（p）$ Explouse Exception e eigeNvalue中恢复所有相关数量的$ p（z）$。我们对这一将军定理的证明在很大程度上依赖于$ \ mathbb {dl}（p）$中的铅笔的代表，该$bézoutiansbybézoutiansby Nakatsukasa，Noferini和Townsend [Siam J. Matrix肛门。应用。 38（1），181-209，2015]。
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 Bézoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181-209, 2015].
