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

Tony F. Chan; Diana C. Resasco

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

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

Tony F. Chan^*, Diana C. Resasco

^*Corresponding author for this work

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

1 Scopus citations

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 - Apr 1986
Externally published	Yes

Bibliographical note

Funding Information:
*The authors were supported in part by the Department of Energy under contract DE-AC02.81ER10996, by the Army Research Office under contract DAAG83-0177, and by a BID-CONICET fellowship from Argentina.

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Geometry and Topology
Numerical Analysis
Algebra and Number Theory

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title = "On the condition of nearly singular matrices under rank-1 perturbations",

abstract = "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 {\^A} in terms of |α| and the angles between z and w and the singular vectors of A corresponding to dn.",

author = "Chan, {Tony F.} and Resasco, {Diana C.}",

note = "Funding Information: *The authors were supported in part by the Department of Energy under contract DE-AC02.81ER10996, by the Army Research Office under contract DAAG83-0177, and by a BID-CONICET fellowship from Argentina.",

year = "1986",

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doi = "10.1016/0024-3795(86)90224-7",

language = "English (US)",

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T1 - On the condition of nearly singular matrices under rank-1 perturbations

AU - Chan, Tony F.

AU - Resasco, Diana C.

N1 - Funding Information: *The authors were supported in part by the Department of Energy under contract DE-AC02.81ER10996, by the Army Research Office under contract DAAG83-0177, and by a BID-CONICET fellowship from Argentina.

PY - 1986/4

Y1 - 1986/4

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

AN - SCOPUS:46149140841

SN - 0024-3795

VL - 76

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EP - 232

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

Abstract

Bibliographical note

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