The resolvent average for positive semidefinite matrices

Abstract

We define a new average -termed the resolvent average -for positive semidefinite matrices. For positive definite matrices, the resolvent average enjoys self-duality and it interpolates between the harmonic and the arithmetic averages, which it approaches when taking appropriate limits. We compare the resolvent average to the geometric mean. Some applications to matrix functions are also given.2000 Mathematics Subject Classification: Primary 47A64; Secondary 15A45, 15A24, 47H05, 90C25.

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

“…, A m ) of [41]. It can be defined as the minimum of a function looking like (15) (with a Bregman distance replacing d R ; see [41,Proposition 2.8]), but it also has a simple, easy-to-compute expression that can be written…”

“…The fact that the resolvent average satisfies (16) comes from techniques of variational analysis (see [41,Theorem 4.8]). …”

On some inequalities for the matrix power and Karcher means

“…Our proof is totally different from that of Lawson and Lim, where the Banach fixed point theorem and the Thompson metric play crucial roles. Some other monotone families of matrix means interpolating between the harmonic mean and the arithmetic mean in terms of resolvent and AGH means appear in [2,11].…”

“…Suppose that X := P t (ω; A) I. By the invariancy of power means under congruence transformation and Hansen's inequality, we have for p ∈[1,2] …”

The resolvent average on symmetric cones

“…This new concept was applied to positive semidefinite matrices under the name of resolvent average [8]. Basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix diagonalization [8].…”