Journal of Inequalities and Applications
Volume 3 (1999), Issue 2, Pages 143-152
doi:10.1155/S1025583499000107
A stronger version of matrix convexity as applied to functions of Hermitian matrices
Department of Mathematics, University of Maryland, College Park 20742, MD, USA
Received 14 January 1998; Revised 17 February 1998
Copyright © 1999 Abram Kagan and Paul J. Smith. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A stronger version of matrix convexity, called hyperconvexity is introduced. It is shown that the function A2 is hyperconvex on the set of Hermitian matrices A and A−1 is hyperconvex on the set of positive definite Hermitian matrices. The new concept makes it possible to consider weighted averages of matrices of different orders. Proofs use properties of the Fisher information matrix, a fundamental concept of mathematical statistics.