The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2 \leqslant r \leqslant kn$ is considered. It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor. For regular matrix polynomials the problem is shown to be equivalent to finding minimal structure preserving perturbations such that a certain block Toeplitz matrix becomes suitably rank deficient. This is then used to characterize the distance via two different optimizations. The first one shows that if $\lambda_0$ is not already an eigenvalue of the matrix polynomial, then the problem is equivalent to computing a generalized notion of a structured singular value. The distance is computed via algorithms like BFGS and Matlab's globalsearch algorithm from the second optimization. Upper and lower bounds of the distance are also derived and numerical experiments are performed to compare them with the computed values of the distance.

中文翻译：

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具有指定初除数的最近矩阵多项式

求出从给定的 $n \times n$ 阶数为 $k$ 的矩阵多项式到具有初等除数 $(\lambda-\lambda_0)^j, \, j \geqslant r 的矩阵多项式集合的距离的问题， $ 用于固定标量 $\lambda_0$ 和 $2 \leqslant r \leqslant kn$ 被考虑。确定不规则的多项式任意接近于具有所需初等因数的规则矩阵多项式。对于正则矩阵多项式，该问题被证明等同于寻找最小结构保留扰动，使得某个块 Toeplitz 矩阵变得适当的秩亏。然后通过两种不同的优化将其用于表征距离。第一个表明如果 $\lambda_0$ 还不是矩阵多项式的特征值，那么这个问题就相当于计算一个结构化奇异值的广义概念。距离是通过 BFGS 和 Matlab 的 globalsearch 算法等算法从第二次优化中计算出来的。还推导出距离的上限和下限，并进行数值实验以将它们与距离的计算值进行比较。