Poset-refined majorization relations
arXiv:2607.28061
The paper refines classical matrix majorization relations by allowing a partial order alignment of eigenvalues or singular values, using LU‑approximation, and applies these refined inequalities to tensor products, symmetric powers, and Kronecker sums.
Abstract
Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.
13 pages