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 -