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.