### 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 |
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Pages (from-to) | 1489-1512 |

Number of pages | 24 |

Journal | Linear Algebra and Its Applications |

Volume | 434 |

Issue number | 6 |

DOIs | |

Publication status | Published - 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

### Cite this

*Linear Algebra and Its Applications*,

*434*(6), 1489-1512. https://doi.org/10.1016/j.laa.2010.11.023