Geometric mean block matrices

We consider an m×m block matrix G with entries Ai#Aj where A1,…,Am are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A1,…,Am is Γ-commuting; it can be transformed to a commuting family of positive definite matrices by a congruence transformation. This result via Γ-commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices.