Perturbation bounds on the extremal singular values of a matrix after appending a column

TL;DR

This paper investigates how the largest and smallest singular values of a matrix change when a new column is added, providing new bounds and applications in signal processing and control theory.

Contribution

It introduces new lower and upper bounds on the perturbation of extremal singular values after appending a column, with simple proofs based on characteristic polynomial analysis.

Findings

New bounds on singular value perturbations are established.

Applications demonstrated in signal processing and control theory.

Comparison of existing bounds with new results.

Abstract

In this paper, we study the perturbation of the extreme singular values of a matrix in the particular case where it is obtained after appending an arbitrary column vector. Such results have many applications in bifurcation theory, signal processing, control theory and many other fields. In the first part of this paper, we review and compare various bounds from recent research papers on this subject. We also present a new lower bound and a new upper bound on the perturbation of the operator norm is provided. Simple proofs are provided, based on the study of the characteristic polynomial rather than on variational methods, as e.g. in \cite{Li-Li}. In a second part of the paper, we present applications to signal processing and control theory.