发表机构
Aberystwyth University; South Ural State University（阿伯里斯特威斯大学; 南乌拉尔国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对矩阵函数的Wiener--Hopf分解，引入P-规范化概念，利用Birkhoff分解和PLU分解，证明任意可逆矩阵函数均可实现唯一规范化分解，并支持误差估计。
AI 中文摘要
对于矩阵函数$A(t)$，我们研究其Wiener--Hopf分解$A(t)=A_-(t)D(t)A_+(t)$的规范化，该规范化保证了因子$A_\pm(t)$的唯一性。此前，这一问题已针对二阶矩阵函数、稳定分解以及具有不同部分指标的矩阵函数的分解进行了研究。解决规范化问题的主要工具是I.C. Gohberg和M.G. Krein关于因子$A_\pm$一般形式的定理、矩阵函数的Birkhoff分解以及数值矩阵$A_-\infty$的块PLU分解。为了解决规范化问题，引入了P-规范化分解的概念。矩阵函数的Birkhoff分解在定义这一概念中起关键作用。置换矩阵$P$由数值矩阵$A_-\infty$的PLU分解确定。我们证明了任何可逆矩阵函数$A(t)$都允许P-规范化。这种规范化保证了Wiener--Hopf分解的唯一性。$A(t)$的P-规范化分解使我们能够找到$A(t)$的Birkhoff分解。没有规范化，就不可能获得分解因子近似计算中绝对误差的显式估计。
英文摘要
{For a matrix function $A(t)$, we study the normalisation of its Wiener--Hopf factorisation $A(t)=A_-(t)D(t)A_+(t)$ which guarantees the uniqueness of the factors $A_\pm(t)$. Previously, this problem was investigated for second-order matrix functions, the stable factorisation, and the factorisation of matrix functions with distinct partial indices. The main tools for solving the problem of normalisation are the theorem of I.C. Gohberg and M.G. Krein on a general form of the factors $A_\pm$, a Birkhoff factorisation of matrix functions, and a block PLU factorisation of the numerical matrix $A_-(\infty)$. To solve the normalisation problem, a concept of P-normalised factorisation is introduced. The Birkhoff factorisation of matrix functions plays a key role in defining this concept. The permutation matrix $P$ is determined by PLU factorisation of the numerical matrix $A_-(\infty)$. We proved that any invertible matrix function $A(t)$ admits P-normalisation. This normalisation guarantees the uniqueness of the Wiener--Hopf factorisation. The P-normalised factorisation of $A(t)$ allows us to find the Birkhoff factorisation of $A(t)$. Without normalisation, it is impossible to obtain explicit estimates of the absolute errors in the approximate computation of the factorisation factors.}
Comments22 pages, without tables and figures
