论文标题
在严格的等效性下，矩阵铅笔的束
On bundles of matrix pencils under strict equivalence
论文作者
论文摘要
基质铅笔（在严格的等效性下）是具有相同Kronecker规范形式的铅笔组，直到特征值（即它们是严格等效性的轨道的无限结合）。 Arnold于1971年引入了1990年代的基质铅笔束概念，遵循相似性的矩阵概念，从那时起就广泛使用了矩阵。尽管有大量的文献致力于描述矩阵铅笔束的拓扑，但在这种情况下，一些相关问题仍然开放。例如，以下两个：（a）为捆绑包（标准拓扑）之间的包含关系提供了一种表征； （b）捆绑包在关闭中是否打开？本文的主要目标是为这两个问题提供明确的答案。为了获得这个答案，我们还审查和/或正式化了文献中已经存在的一些概念和结果。我们还证明，在相似性下的矩阵捆绑包以及矩阵多项式的捆绑（定义为具有相同频谱信息的相同等级的$ M \ times n $矩阵多项式的集合，直至特征值）在其关闭中是打开的。
Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of $m\times n$ matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure.
