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.

翻译：矩阵多项式束是一组具有相同尺寸、次数且除特征值具体数值外具有相同特征结构的矩阵多项式。已知铅笔 $L$（即次数为1的矩阵多项式）的束闭包 $\mathcal{B}(L)$ 是 $\mathcal{B}(L)$ 自身与有限个其他束的并集。本文的第一个主要贡献是证明这些束的维数严格小于 $\mathcal{B}(L)$ 的维数。第二个主要贡献是证明次数大于1的矩阵多项式的束闭包同样是该束自身与有限个维数更小的其他束的并集。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，推导了矩阵铅笔束的（余）维数公式，并刻画了同尺寸同次数矩阵多项式束闭包之间的包含关系。