A Matrix Generalization of the Hardy-Littlewood-Pólya Rearrangement Inequality and Its Applications
Man-Chung Yue
TL;DR
This work develops a matrix analogue of the Hardy-Littlewood-Pólya rearrangement inequality by solving an optimization over orthogonal matrices for positive definite $A,B$, under a differentiability condition on $f$ with $s\mapsto s f'(s)$ monotone. The main result yields inequalities $\sum_i f(\lambda_i(A)\lambda_{n-i+1}(B)) \le \sum_i f(\lambda_i(B^{1/2}AB^{1/2})) \le \sum_i f(\lambda_i(A)\lambda_i(B))$, with the middle term equal to $\mathrm{Tr}(f(B^{1/2}AB^{1/2}))$, and extends to rectangular matrices with a PSD extension when $f$ is right-continuous at $0$. Leveraging this, the paper derives high-signal inequalities for Schatten quasi-norms, affine-invariant distances on the manifold $\mathbb{P}_n$, and Alpha-Beta log-determinant divergences, all expressed in terms of eigenvalue or singular value spectra. The approach reveals a commutation principle at optimality and suggests future extensions to Euclidean Jordan algebras, broadening the impact of rearrangement ideas in matrix analysis and information geometry.
Abstract
By analyzing an optimization problem over orthogonal matrices, we prove a generalization of the Hardy-Littlewood-Pólya 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.