A stronger version of matrix convexity as applied to functions of Hermitian matrices (Q1288853)

It is known that the function \(f(A) = A^2\) is matrix convex on the set \(H_s\) of Hermitian \(s \times s\) matrices and \(g(A) = A^{-1}\) is matrix convex on the set \(H^+_s\) of positive definite Hermitian \(s \times s\) matrices. The authors have recently suggested 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, in contrast to the known analytical proofs. In this paper the authors introduce a stronger notion of matrix convexity, called hyperconvexity. In the new notion of hyperconvexity, convex combinations are replaced by weighted averages by matrices of different order. The authors show that the above functions \(f\) and \(g\) are hyperconvex on the sets \(\bigcup^\infty_{s = 1} H_s\) and \(\bigcup^\infty_{s = 1} H_s^+\), respectively. The proofs are based on basic properties of the Fisher information matrix for normal random vectors. The authors also give analytical proofs of their new results.