Trace-Inequalities and Matrix-Convex Functions

Abstract

A real-valued continuous function f(t) on an interval (α,β) gives rise to a map X↦f(X) via functional calculus from the convex set of n×n Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: Tr(f(B)−f(A))(C−B)≤Tr(f(C)−f(B))(B−A) for A≤B≤C. A related topic will be also discussed.