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

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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

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