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

2014/06/20 by Stéphane Chrétien, Chretien, Stephane, Sébastien Darses +1 · 2 citations
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Sparse and Compressive Sensing Techniques #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1406.5441

openalex publication_date 2014/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citeLi-Li. In a second part of the paper, we present applications to signal processing and control theory.

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