Positiveness for the Riemannian geodesic block matrix
Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 917-925
Published online July 2, 2020
Printed July 31, 2020
Jinmi Hwang, Sejong Kim
Chungbuk National University; Chungbuk National University
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.
Keywords : Riemannian geodesic, Loewner order, Schur complement, stochastic matrix
MSC numbers : 15A42, 15A18
Supported by : This research was supported by Chungbuk National University (2019)
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