The Resolvent Average for Positive Semidefinite Matrices

TL;DR

This paper introduces the resolvent average, a novel mean for positive semidefinite matrices, which interpolates between harmonic and arithmetic averages and exhibits self-duality, with applications to matrix functions.

Contribution

The paper defines the resolvent average for positive semidefinite matrices and explores its properties, including self-duality and interpolation behavior, comparing it to existing means.

Findings

Resolvant average interpolates between harmonic and arithmetic means.

Resolvant average exhibits self-duality property.

Applications to matrix functions demonstrated.

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.