Factorizations and geometric means of positive definite matrices

In this paper we provide a new class of (metric) geometric means of positive definite matrices varying over Hermitian unitary matrices. We show that each Hermitian unitary matrix induces a factorization of the cone Pm of m×m positive definite Hermitian matrices into geodesically convex subsets and a Hadamard metric structure on Pm. An explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two-variable mean is different to the standard geometric mean. Some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied.