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arXiv 2102.09726math.NAcs.NA
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矩阵多项式线性化的等价性

Equivalences for Linearizations of Matrix Polynomials

Robert M. Corless, Leili Rafiee Sevyeri, B. David Saunders

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摘要：

计算次数至多为$z$中的$\ell$的矩阵多项式${\bf P}(z) \in \mathbb{C}^{n\times n}[z]$（简称为等级$\ell$）的特征值的一种有用的标准方法是首先将${\bf P}(z)$变换为等价的线性矩阵多项式${\bf L}(z)=z{\bf B}-{\bf A}$，称为伴随束，其中${\bf A}$和${\bf B}$通常具有比${\bf P}(z)$更大的维数，但是${\bf L}(z)$现在仅为$z$中的等级$1$。${\bf L}(z)$的本征值和本征向量可以通过例如QZ算法进行数值计算。如果${\bf L}(z)$是所谓的${\bf P}(z)$的"强线性化"，则${\bf P}(z)$的本征向量，包括对于无限本征值的本征向量，也可以从${\bf L}(z)$的本征向量恢复。在本文中，我们展示了如何使用算法计算的一个标量多项式的同伴矩阵的Hermite规范形，以指导发现么模矩阵多项式的余因子${\bf E}(z)$和${\bf F}(z)$，通过方程${\bf E}(z){\bf L}(z){\bf F}(z) = \mathrm{diag}( {\bf P}(z), {\bf I}_n, \ldots, {\bf I}_n)$，明确显示的等价性${\bf P}(z)$和${\bf L}(z)$。通过这种方法，我们给出了新的显式结构的几个线性化使用不同的多项式基。我们将这些新的么模对与严格等价构造的么模对进行对比，其中一些对本文也是新的。我们讨论了这个实验的局限性，计算发现的方法找到么模余因子。

英文摘要：

One useful standard method to compute eigenvalues of matrix polynomials ${\bf P}(z) \in \mathbb{C}^{n\times n}[z]$ of degree at most $\ell$ in $z$ (denoted of grade $\ell$, for short) is to first transform ${\bf P}(z)$ to an equivalent linear matrix polynomial ${\bf L}(z)=z{\bf B}-{\bf A}$, called a companion pencil, where ${\bf A}$ and ${\bf B}$ are usually of larger dimension than ${\bf P}(z)$ but ${\bf L}(z)$ is now only of grade $1$ in $z$. The eigenvalues and eigenvectors of ${\bf L}(z)$ can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of ${\bf P}(z)$, including those for infinite eigenvalues, can also be recovered from eigenvectors of ${\bf L}(z)$ if ${\bf L}(z)$ is what is called a "strong linearization" of ${\bf P}(z)$. In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors ${\bf E}(z)$ and ${\bf F}(z)$ which, via the equation ${\bf E}(z){\bf L}(z){\bf F}(z) = \mathrm{diag}( {\bf P}(z), {\bf I}_n, \ldots, {\bf I}_n)$, explicitly show the equivalence of ${\bf P}(z)$ and ${\bf L}(z)$. By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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