Monotonicity of the matrix geometric mean

Abstract

An attractive candidate for the geometric mean of m positive definite matrices A₁, . . . , A_m is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by Lawson and Lim. We give an elementary proof of this property using standard matrix analysis and some counting arguments. We derive some new inequalities for G. One of these says that, for any unitarily invariant norm, ||| G ||| is not bigger than the geometric mean of |||A ₁|||, . . . , |||A _m |||.