Matrix-valued Jensen's inequality and its application to eigenvalue analysis

We first establish Jensen's inequality for lower semicontinuous convex functions from a Hadamard space into an “ordered” Hadamard space. We apply Jensen's inequality to matrix valued integrable functions on probability measure spaces and provide new results on matrix analysis and eigenvalue analysis by discovering a variety of Lipschitz convex functions related to eigenvalue maps and Löwner ordering of positive definite matrices.