Convexity of the inverse and Moore-Penrose inverse

Kenneth Nordström

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.

Original languageEnglish
Pages (from-to)1489-1512
Number of pages24
JournalLinear Algebra and Its Applications
Volume434
Issue number6
DOIs
Publication statusPublished - Mar 15 2011
Externally publishedYes

Fingerprint

Moore-Penrose Inverse
Convexity
Positive definite
Non-negative
Strict Convexity
Reproducing Kernel Hilbert Space
Partial ordering
Null Space
Positive definite matrix
Matrix Function
Matrix Inequality
Hilbert spaces
Concepts

Keywords

  • Extremal representation
  • Indefinite inner product
  • Matrix mean
  • Reproducing kernel Hilbert space
  • Strongly convex matrix function

ASJC Scopus subject areas

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

Cite this

Convexity of the inverse and Moore-Penrose inverse. / Nordström, Kenneth.

In: Linear Algebra and Its Applications, Vol. 434, No. 6, 15.03.2011, p. 1489-1512.

Research output: Contribution to journalArticle

Nordström, Kenneth. / Convexity of the inverse and Moore-Penrose inverse. In: Linear Algebra and Its Applications. 2011 ; Vol. 434, No. 6. pp. 1489-1512.
@article{d14c516655d24971bcabb41532755a51,
title = "Convexity of the inverse and Moore-Penrose inverse",
abstract = "Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.",
keywords = "Extremal representation, Indefinite inner product, Matrix mean, Reproducing kernel Hilbert space, Strongly convex matrix function",
author = "Kenneth Nordstr{\"o}m",
year = "2011",
month = "3",
day = "15",
doi = "10.1016/j.laa.2010.11.023",
language = "English",
volume = "434",
pages = "1489--1512",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier",
number = "6",

}

TY - JOUR

T1 - Convexity of the inverse and Moore-Penrose inverse

AU - Nordström, Kenneth

PY - 2011/3/15

Y1 - 2011/3/15

N2 - Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.

AB - Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.

KW - Extremal representation

KW - Indefinite inner product

KW - Matrix mean

KW - Reproducing kernel Hilbert space

KW - Strongly convex matrix function

UR - http://www.scopus.com/inward/record.url?scp=79551682426&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79551682426&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.11.023

DO - 10.1016/j.laa.2010.11.023

M3 - Article

VL - 434

SP - 1489

EP - 1512

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 6

ER -