Matrix power means and the Karcher mean

Abstract

We define a new family of matrix means { P t ( ω ; A ) } t ∈ [ − 1 , 1 ] , where ω and A vary over all positive probability vectors in R n and n-tuples of positive definite matrices resp. Each of these means except t ≠ 0 arises as a unique positive definite solution of a non-linear matrix equation, satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means. The main result is that the Karcher mean coincides with the limit of power means as t → 0 . This provides not only a sequence of matrix means converging to the Karcher mean, but also a simple proof of the monotonicity of the Karcher mean, conjectured by Bhatia and Holbrook, and other new properties, which have recently been established by Lawson and Lim and also Bhatia and Karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric.