矩阵多项式块最小基$\ell$-化
Block minimal bases $\ell$-ifications of matrix polynomials
AI总结：
本文提出一种构造矩阵多项式强$\ell$-化的新型方法，利用块最小基矩阵多项式性质，适用于$\ell < d$且$\ell$整除$nd$或$md$的情况，保证数值应用中的稳定性与可恢复性。
摘要：
解决矩阵多项式相关特征值问题的标准方法是将多项式系数嵌入矩阵 pencil，即强线性化。然而在某些情况下，用低阶矩阵多项式替代线性化更方便。本文提出强$\ell$-化概念，即构造一个度数为$\ell$的矩阵多项式，具有相同的有限和无限初等因子及左右最小指数。本文提出一种新方法，用于构造大小为$m\times n$、等级为$d$的矩阵多项式强$\ell$-化，当$\ell < d$且$\ell$整除$nd$或$md$时。该方法基于称为
英文摘要：
The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong $\ell$-ification of a matrix polynomial, which is a matrix polynomial of degree $\ell$ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong $\ell$-ifications of matrix polynomials of size $m\times n$ and grade $d$ when $\ell< d$, and $\ell$ divides $nd$ or $md$. This method is based on a family called "strong block minimal bases matrix polynomials", and relies heavily on properties of dual minimal bases. We show how strong block minimal bases $\ell$-ifications can be constructed from the coefficients of a given matrix polynomial $P(λ)$. We also show that these $\ell$-ifications satisfy many desirable properties for numerical applications: they are strong $\ell$-ifications regardless of whether $P(λ)$ is regular or singular, the minimal indices of the $\ell$-ifications are related to those of $P(λ)$ via constant uniform shifts, and eigenvectors and minimal bases of $P(λ)$ can be recovered from those of any of the strong block minimal bases $\ell$-ifications. In the special case where $\ell$ divides $d$, we introduce a subfamily of strong block minimal bases matrix polynomials named "block Kronecker matrix polynomials", which is shown to be a fruitful source of companion $\ell$-ifications.
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