Short note on the perturbation of operators with dyadic products

Abstract

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 $m\geq k$.