We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization.
Geometric mean flows and the Cartan barycenter on the Wasserstein space over positive definite matrices
Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter
Local extrema for Procustes problems in the set of positive definite matrices
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