New multivariable mean from nonlinear matrix equation associated to the harmonic mean

Abstract

Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and R\'{e}nyi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation: $$ X = \left[ \sum_{i=1}^{n} w_{i} [ (1-t) X + t A_{i} ]^{-1} \right]^{-1}, $$ where $t \in (0,1]$. We prove that this equation has a unique solution and define a new mean, which we denote as $G_{t}(\omega; \mathbb{A})$. We explore important properties of the mean $G_{t}(\omega; \mathbb{A})$ including the relationship with matrix power mean, and show that the mean $G_{t}(\omega; \mathbb{A})$ is monotone in the parameter $t$. Finally, we connect the mean $G_{t}(\omega; \mathbb{A})$ to a barycenter for the log-determinant divergence.