A Review of Geometric Mean of Positive Definite Matrices

In the paper [1], Pusz and Woronowicz first gave the geometric mean of two positive definite matrices. This mean has similar properties to those of the geometric mean of two positive numbers. In [2], Ando, Li and Mathias listed ten properties that a geometric mean of m positive definite matrices should satisfy. Then gave a definition of geometric mean of m matrices by a iteration which satisfies these ten properties. For the geometric mean of two positive definite matrices, there is an interesting relationship between matrix geometric mean and the information metric. Consider the set of all positive matrices as a Riemannian manifold with the information metric. Then the geometric mean of two matrices in the manifold is just the middle point of the geodesic connecting them. In this paper, we review this notion and present two different proofs, the variation method