Convexity of the inverse and Moore-Penrose inverse

Kenneth Nordström

doi:10.1016/j.laa.2010.11.023

Convexity of the inverse and Moore-Penrose inverse

Kenneth Nordström

Department of Mathematics

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

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 language	English
Pages (from-to)	1489-1512
Number of pages	24
Journal	Linear Algebra and Its Applications
Volume	434
Issue number	6
DOIs	https://doi.org/10.1016/j.laa.2010.11.023
Publication status	Published - Mar 15 2011

Keywords

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

ASJC Scopus subject areas

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

Access to Document

10.1016/j.laa.2010.11.023

Fingerprint Dive into the research topics of 'Convexity of the inverse and Moore-Penrose inverse'. Together they form a unique fingerprint.

Cite this

@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",

note = "Funding Information: This work was completed while visiting the Department of Mathematical Sciences, Montana State University. I am grateful to the Head of the Department Ken Bowers for generous provision of facilities, and to the Finnish Society of Sciences and Letters for a research grant. Thanks are also due to Beth Ruskai and Richard Brualdi for helpful comments on the manuscript. E-mail address: kenneth.nordstrom@oulu.fi 0024-3795/$ - see front matter {\textcopyright} 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.11.023",

year = "2011",

month = mar,

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

N1 - Funding Information: This work was completed while visiting the Department of Mathematical Sciences, Montana State University. I am grateful to the Head of the Department Ken Bowers for generous provision of facilities, and to the Finnish Society of Sciences and Letters for a research grant. Thanks are also due to Beth Ruskai and Richard Brualdi for helpful comments on the manuscript. E-mail address: kenneth.nordstrom@oulu.fi 0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.11.023

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

AN - SCOPUS:79551682426

VL - 434

SP - 1489

EP - 1512

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 6

ER -