论文标题
多项式和理性矩阵的根矢量：理论与计算
Root vectors of polynomial and rational matrices: theory and computation
论文作者
论文摘要
多项式矩阵$ p（λ）$的根多项式概念在[f中进行了彻底研究。 Dopico和V. noferini，根多项式及其在基质多项式理论中的作用，线性代数应用。 584：37--78，2020]。在本文中，我们将这种系统的方法扩展到通用理性矩阵$ r（λ）$，可能是单数，并且可能具有结合杆/零对。我们在任意领域中讨论具有系数的有理矩阵的相关理论。作为副产品，我们获得了理性矩阵$ r（λ）$的特征值和特征向量的明智定义，而无需假设$ r（λ）$具有完整的列等级或特征值也不是极点。然后，我们专门研究复杂的领域，并提供一种实用的算法来计算它们，这是基于对理性矩阵$ r（λ）$的最小状态空间实现的构建，然后在线性化铅笔上使用楼梯算法来计算无效空间以及在给定点$λ__0$λ__0$上的dolonomials。如果$λ_0$也是杆子，那么有必要应用一个预处理步骤，该步骤可以消除杆子，同时可以恢复原始矩阵的根矢量：在这种情况下，我们同时研究相关的理论（一般字段）和算法实现（在复数上），基于最小的状态空间，仍然基于最小的状态空间。
The notion of root polynomials of a polynomial matrix $P(λ)$ was thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020]. In this paper, we extend such a systematic approach to general rational matrices $R(λ)$, possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix $R(λ)$, without any need to assume that $R(λ)$ has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix $R(λ)$ and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point $λ_0$. If $λ_0$ is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.
