关于并矢积算子扰动的注记
Short note on the perturbation of operators with dyadic products
摘要：
在本文中，我们使用抽象向量空间和它们的无规范基。我们的一些结果也可以推广到无限维向量空间，但这里我们只考虑有限维空间。我们专注于一般扰动问题。假设$B:V\to V$是线性算子，其被扰动为$B'=B+Q$。我们研究的问题如何行列式和逆的变化，因为这种扰动。在我们的方法中，运营商$Q$是作为一个和并积$Q=\sum_{i=1}^{k}v_{i}\otimes p_{i}$，其中$v_{i}\in V$和$p_{i}\in V^{*}$。本文导出了$\det B'$和$(B')^{-1}$的一个$m$-阶（$m\in\mathbb{N}$）近似公式，它给出了当$m\geq k$ .
英文摘要：
In this paper we use abstract vector spaces and their duals without any canonical basis. Some of our results can be extended to infinite dimensional vector spaces too, but here we consider only finite dimensional spaces. We focus on a general perturbation problem. Assume that $B:V\to V$ is a linear operator, which is perturbated to $B'=B+Q$. We examine the question how the determinant and the inverse change, because of this perturbation. In our approach the operator $Q$ is given as a sum of dyadic products $Q=\sum_{i=1}^{k}v_{i}\otimes p_{i}$, where $v_{i}\in V$ and $p_{i}\in V^{*}$. In this paper we derive an $m$-th order ($m\in\mathbb{N}$) approximation formula for $\det B'$ and $(B')^{-1}$, which gives the exact result if $m\geq k$.
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。
