Positiveness for the Riemannian geodesic block matrix

Abstract

It has been shown that the geometric mean $A \# B$ of positive definite Hermitian matrices $A$ and $B$ is the maximal element $X$ of Hermitian matrices such that \begin{displaymath} \left( \begin{array}{cc} A & X \\ X & B \\ \end{array} \right) \end{displaymath} is positive semi-definite. As an extension of this result for the $2 \times 2$ block matrix, we consider in this article the block matrix $[[ A \#_{w_{ij}} B ]]$ whose $(i,j)$ block is given by the Riemannian geodesics of positive definite Hermitian matrices $A$ and $B$, where $w_{ij} \in \mathbb{R}$ for all $1 \leq i, j \leq m$. Under certain assumption of the Loewner order for $A$ and $B$, we establish the equivalent condition for the parameter matrix $\omega = [w_{ij}]$ such that the block matrix $[[ A \#_{w_{ij}} B ]]$ is positive semi-definite.