Factorizations and geometric means of positive definite matrices (Q448363)

The author presents a new class of (metric) geometric means of positive definite matrices varying over Hermitian unitary matrices. He proves that each Hermitian unitary matrix induces a factorization of the cone \(\mathbb P_{m}\) of \(m\times m\) positive definite Hermitian matrices into geodesically convex subsets and a Hadamard metric structure on \(\mathbb P_{m}\). He also gives an explicit formula for the corresponding metric midpoint operation in terms of the geometric and spectral geometric means and shows 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 a finite number of positive definite matrices are also studied.