### 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 | |

Publication status | Published - Mar 15 2011 |

Externally published | Yes |

### Fingerprint

### 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

*Linear Algebra and Its Applications*,

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

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 434, no. 6, pp. 1489-1512. https://doi.org/10.1016/j.laa.2010.11.023

}

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 -