关于矩阵束与矩阵多项式的丛闭包
On bundle closures of matrix pencils and matrix polynomials
AI总结：
本文研究矩阵多项式丛的闭包结构，证明矩阵束及高次矩阵多项式的丛闭包均为原丛与有限多个维数更小的丛之并，并给出丛维数公式和闭包包含关系的刻画。
摘要：
矩阵多项式丛是指具有相同规模与次数、且除特征值具体取值外具有相同特征结构的矩阵多项式集合。已知矩阵束 $L$（即次数为 $1$ 的矩阵多项式）的丛闭包 $\mathcal{B}(L)$ 是 $\mathcal{B}(L)$ 本身与有限多个其他丛的并集。本文的第一个主要贡献是证明其中每个丛的维数都严格小于 $\mathcal{B}(L)$ 的维数。第二个主要贡献是证明次数大于 $1$ 的矩阵多项式的丛闭包同样是该丛本身与有限多个维数更小的其他丛的并集。为得到这些结果，我们利用特征值部分重数的 Weyr 特征以及（左、右）最小指标，给出了矩阵束丛的（余）维数公式，并给出了相同规模与次数的两个矩阵多项式丛的闭包之间包含关系的刻画。
英文摘要：
Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The first main contribution of this paper is to prove that the dimension of each of these bundles is strictly smaller than the dimension of $\mathcal{B}(L)$. The second main contribution is to prove that also the closure of the bundle of a matrix polynomial of grade larger than 1 is the union of the bundle itself with a finite number of other bundles of smaller dimension. To get these results we obtain a formula for the (co)dimension of the bundle of a matrix pencil in terms of the Weyr characteristics of the partial multiplicities of the eigenvalues and of the (left and right) minimal indices, and we provide a characterization for the inclusion relationship between the closures of two bundles of matrix polynomials of the same size and grade.
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