- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 Positiveness for the Riemannian geodesic block matrix Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 917-925 https://doi.org/10.4134/CKMS.c200033Published online July 5, 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) Downloads: Full-text PDF   Full-text HTML