Local Lidskii's theorems for unitarily invariant norms

TL;DR

This paper proves that Lidskii's inequalities characterize all global minimizers of unitarily invariant norm functions on matrix orbits, and shows local minimizers are also global, with applications to finite frame theory.

Contribution

It establishes that Lidskii's inequalities fully describe global minimizers and that local minimizers are globally optimal, extending understanding of matrix optimization problems.

Findings

Lidskii's inequalities characterize all global minimizers.

Local minimizers are also global minimizers.

Results apply to finite frame operator distances.

Abstract

Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory.