A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

矩阵多项式的许多理论和计算问题通过传递到线性化来解决。因此，需要一种将线性化中的扰动与相应矩阵多项式中的等效扰动联系起来的扰动理论。在本文中，我们开发了一种算法，可以找到矩阵多项式的矩阵系数的哪个扰动对应于整个线性化铅笔的给定扰动。此外，我们找到了通过严格等价将线性化的扰动转换为扰动多项式的线性化的变换矩阵。为简单起见，我们展示了第一个伴随线性化的结果，但它们可以推广到更广泛的线性化类别。