Averaging Over the Unitarian Group and the Monotonicity Conjecture of Merris and Watkins

TL;DR

This paper proves that the monotonicity conjecture of Merris and Watkins holds on average for matrices with fixed spectra when sampled uniformly from the unitary group.

Contribution

It demonstrates the average validity of the monotonicity conjecture using Haar measure on the unitary group, a novel probabilistic approach.

Findings

The conjecture holds on average over the unitary group.

Haar measure provides a natural probability space for matrices.

Results support the conjecture's validity in a probabilistic sense.

Abstract

We show that the monotonicity conjecture of Merriss and Watkins is true in average when taking the set of matrices of given non negative spectra as probability space with respect to the Haar measure of the unitarian group .