Short note on the perturbation of operators with dyadic products

TL;DR

This paper develops a mathematical framework to analyze how determinants and inverses of linear operators change under perturbations expressed as sums of dyadic products, providing approximation formulas with exactness conditions.

Contribution

It introduces a new approach to perturbation analysis of operators using dyadic product sums, with explicit formulas for determinant and inverse changes.

Findings

Derived m-th order approximation formulas for determinants and inverses

Formulas become exact when the order m exceeds the number of dyadic terms

Applicable to finite-dimensional vector spaces, extendable to infinite dimensions

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 $m\geq k$.