Abstract
A stronger version of matrix convexity, called hyperconvexity is introduced. It is shown that the function A 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.
INTRODUCTION
It is known that the functionf(A)= A 2 is matrix convex on the set of Hermitian s s matrices and fz(A) A -is matrix convex on the set 7-/ + of positive definite Hermitian s s matrices. For analytical proofs of these and related results see [4, Ch. 16].
INTRODUCTION
The authors have recently suggested in [3] two purely statistical proofs of the above results, one based on the Gauss-Markov theorem for least squares estimators and the other based on properties of the Fisher information matrix. Thus, x O, the null vector is Rn.
INTRODUCTION
Proofof Theorem 2 The idea is the same as in the proof of Theorem 1, though the choice of X1,..., X is different.
INTRODUCTION
Namely, let X. be an s-variate normal random vector with E X. BO and V(Xj)= A. The probability density function of X. is which is exactly (2.10).

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