Journal of the
Korean Mathematical Society JKMS

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.

Keywords: Positive matrix completion, matrix geometric mean, Schur complement

MSC numbers: Primary 47A63, 47A64, 15A83, 15B48

Supported by: This work of S. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1C1B6001394).