On the condition of nearly singular matrices under rank-1 perturbations

Tony Chan; Diana C. Resasco

doi:10.1016/0024-3795(86)90224-7

On the condition of nearly singular matrices under rank-1 perturbations

Tony Chan^*, Diana C. Resasco

^*Corresponding author for this work

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

Let A be an n-by-n nearly singular matrix with Rank(A)≥n - 1 and singular values d₁≥⋯≥d_n-1 > d_n, where d_n can be small or zero. Consider the rank-1 modification of A: A ̂=A+αzw^T, with {norm of matrix}z{norm of matrix} = {norm of matrix}w{norm of matrix} = 1. We give lower and upper bounds for the condition number of Â in terms of |α| and the angles between z and w and the singular vectors of A corresponding to d_n.

Original language	English (US)
Pages (from-to)	223-232
Number of pages	10
Journal	Linear Algebra and Its Applications
Volume	76
Issue number	C
DOIs	https://doi.org/10.1016/0024-3795(86)90224-7
State	Published - Jan 1 1986

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis

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AU - Resasco, Diana C.

PY - 1986/1/1

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N2 - Let A be an n-by-n nearly singular matrix with Rank(A)≥n - 1 and singular values d1≥⋯≥dn-1 > dn, where dn can be small or zero. Consider the rank-1 modification of A: A ̂=A+αzwT, with {norm of matrix}z{norm of matrix} = {norm of matrix}w{norm of matrix} = 1. We give lower and upper bounds for the condition number of Â in terms of |α| and the angles between z and w and the singular vectors of A corresponding to dn.

AB - Let A be an n-by-n nearly singular matrix with Rank(A)≥n - 1 and singular values d1≥⋯≥dn-1 > dn, where dn can be small or zero. Consider the rank-1 modification of A: A ̂=A+αzwT, with {norm of matrix}z{norm of matrix} = {norm of matrix}w{norm of matrix} = 1. We give lower and upper bounds for the condition number of Â in terms of |α| and the angles between z and w and the singular vectors of A corresponding to dn.

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U2 - 10.1016/0024-3795(86)90224-7

DO - 10.1016/0024-3795(86)90224-7

M3 - Article

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - C

ER -

On the condition of nearly singular matrices under rank-1 perturbations

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