A Matrix Generalization of the Hardy-Littlewood-P\'olya Rearrangement Inequality and Its Applications

By analyzing an optimization problem over orthogonal matrices, we prove a generalization of the Hardy-Littlewood-P'olya rearrangement inequality to positive definite matrices. The inequality is then extended to rectangular matrices. Using our main results, we derive new inequalities for several distance-like functions encountered in various signal processing or machine learning applications.