We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.

中文翻译：

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矩阵多项式空间的几何

我们通过构造矩阵多项式的轨道和束的闭合层次图（分层）来研究通用矩阵多项式的小扰动如何改变其基本除数和最小指数。为了解决这个问题，我们为矩阵多项式的第一个伴随Fiedler线性化构造了分层图。回想一下，第一个伴随的Fiedler线性化以及所有Fiedler线性化都是具有特定块结构的矩阵铅笔。此外，我们表明分层图不取决于Fiedler线性化的选择，这意味着矩阵多项式Fiedler线性化的所有空间都具有相同的几何形状（拓扑）。该几何形状与矩阵多项式空间的几何形状重合。