A note on the convexity of the Moore–Penrose inverse

Kenneth Nordstrom Olof

Research output: Contribution to journalArticle

Abstract

This note is a sequel to an earlier study (Nordström [7]) on convexity properties of the inverse and Moore–Penrose inverse, in which the following question was raised. Given nonnegative definite matrices A and B with Moore–Penrose inverses A+ and B+, respectively, can one show that (λA+λ‾B)+⩽λA++λ‾B+ holding for a single λ∈]0,1[is enough to guarantee its validity for all λ∈]0,1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and λ‾:=1−λ.) In this note an affirmative answer is provided to this question.

Original languageEnglish
Pages (from-to)143-148
Number of pages6
JournalLinear Algebra and Its Applications
Volume538
DOIs
Publication statusPublished - Feb 1 2018

Fingerprint

Moore-Penrose Inverse
Convexity
Non-negative
Partial ordering
Convex Cone
Cones

Keywords

  • Generalized inverse
  • Jensen convexity
  • Loewner ordering
  • Midpoint convexity

ASJC Scopus subject areas

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

Cite this

A note on the convexity of the Moore–Penrose inverse. / Nordstrom Olof, Kenneth.

In: Linear Algebra and Its Applications, Vol. 538, 01.02.2018, p. 143-148.

Research output: Contribution to journalArticle

Nordstrom Olof, Kenneth. / A note on the convexity of the Moore–Penrose inverse. In: Linear Algebra and Its Applications. 2018 ; Vol. 538. pp. 143-148.
@article{59634571c78c452ca90722b572a13ba4,
title = "A note on the convexity of the Moore–Penrose inverse",
abstract = "This note is a sequel to an earlier study (Nordstr{\"o}m [7]) on convexity properties of the inverse and Moore–Penrose inverse, in which the following question was raised. Given nonnegative definite matrices A and B with Moore–Penrose inverses A+ and B+, respectively, can one show that (λA+λ‾B)+⩽λA++λ‾B+ holding for a single λ∈]0,1[is enough to guarantee its validity for all λ∈]0,1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and λ‾:=1−λ.) In this note an affirmative answer is provided to this question.",
keywords = "Generalized inverse, Jensen convexity, Loewner ordering, Midpoint convexity",
author = "{Nordstrom Olof}, Kenneth",
year = "2018",
month = "2",
day = "1",
doi = "10.1016/j.laa.2017.10.016",
language = "English",
volume = "538",
pages = "143--148",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier",

}

TY - JOUR

T1 - A note on the convexity of the Moore–Penrose inverse

AU - Nordstrom Olof, Kenneth

PY - 2018/2/1

Y1 - 2018/2/1

N2 - This note is a sequel to an earlier study (Nordström [7]) on convexity properties of the inverse and Moore–Penrose inverse, in which the following question was raised. Given nonnegative definite matrices A and B with Moore–Penrose inverses A+ and B+, respectively, can one show that (λA+λ‾B)+⩽λA++λ‾B+ holding for a single λ∈]0,1[is enough to guarantee its validity for all λ∈]0,1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and λ‾:=1−λ.) In this note an affirmative answer is provided to this question.

AB - This note is a sequel to an earlier study (Nordström [7]) on convexity properties of the inverse and Moore–Penrose inverse, in which the following question was raised. Given nonnegative definite matrices A and B with Moore–Penrose inverses A+ and B+, respectively, can one show that (λA+λ‾B)+⩽λA++λ‾B+ holding for a single λ∈]0,1[is enough to guarantee its validity for all λ∈]0,1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and λ‾:=1−λ.) In this note an affirmative answer is provided to this question.

KW - Generalized inverse

KW - Jensen convexity

KW - Loewner ordering

KW - Midpoint convexity

UR - http://www.scopus.com/inward/record.url?scp=85032037318&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032037318&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2017.10.016

DO - 10.1016/j.laa.2017.10.016

M3 - Article

VL - 538

SP - 143

EP - 148

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -