Extension of block matrix representation of the geometric mean

J. Korean Math. Soc. Published online August 22, 2019

Hana Choi, Hayoung Choi, Sejong Kim, and Hosoo Lee
Sungkyunkwan University, ShanghaiTech University, Chungbuk National Universit, Jeju National University

Abstract : To extend the well-known extremal characterization of the geometric mean of two $n \times n$ positive definite matrices $A$ and $B$, we solve the following problem.
\begin{equation*}
\max \Bigg\{ X:X=X^*,~
\begin{pmatrix}
A & V & X \\
V & B & W \\
X & W & C
\end{pmatrix}
\geq 0
\Bigg \}
\end{equation*}
We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.