J. Korean Math. Soc. 57 (2020), No. 3, pp. 641–653 https://doi.org/10.4134/JKMS.j190272 pISSN: 0304-9914 / eISSN: 2234-3008 EXTENSION OF BLOCK MATRIX REPRESENTATION OF THE GEOMETRIC MEAN Hana Choi, Hayoung Choi, Sejong Kim, and Hosoo Lee Abstract. To extend the well-known extremal characterization of the geometric mean of two n × n positive definite matrices A and B, we solve the following problem: max  X : X = X ∗ ,   A V X V B W X W C   ≥ 0  . We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements. 1. Introduction The geometric mean of two n × n positive definite matrices A and B is given by an explicit formula [1,16]: A#B := A 1 2 (A − 1 2 BA − 1 2 ) 1 2 A 1 2 . It has an extremal characterization as follows: (1) A#B = max  X : X = X ∗ ,  A X X B  ≥ 0  . Here ≤ denotes the Loewner ordering between Hermitian matrices. A multivariable extension of this characterization arises naturally accord- ing to recent developments of multivariable geometric means on the Cartan- Hadamard-Riemannian manifold of positive definite matrices [2,5,6]. However, the extremal characterization of the geometric mean of two positive definite ma- trices does not seem to be easily generalized to multivariable geometric means. An extremal characterization of geometric mean of three positive definite ma- trices are still open in the context of matrix analysis. Received April 9, 2019; Accepted July 25, 2019. 2010 Mathematics Subject Classification. Primary 47A63, 47A64, 15A83, 15B48. Key words and phrases. Positive matrix completion, matrix geometric mean, Schur complement. 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). c 2020 Korean Mathematical Society 641