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 |
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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
- 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 journal › Article
}
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
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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 -