On the distance from a matrix to nilpotents

TL;DR

This paper establishes a lower bound on the distance from a complex matrix to nilpotent matrices under certain conditions, confirming conjectures by MacDonald and Herrero.

Contribution

It proves a new lower bound for the distance to nilpotents for matrices with specific projections, verifying two longstanding conjectures.

Findings

Lower bound of 0.5*sec(pi/(n+2)) for the distance to nilpotents

Verification of MacDonald's 1995 conjecture in special cases

Confirmation of a related conjecture in Herrero's book

Abstract

We prove that the distance from an $n\times n$ complex matrix $M$ to the set of nilpotents is at least $\frac{1}{2}\sec\frac{\pi}{n+2}$ if there is a nonzero projection $P$ such that $PMP=M$ and $M^*M\geq P$. In the particular case where $M$ equals $P$, this verifies a conjecture by G.W. MacDonald in 1995. We also confirm a related conjecture in D.A. Herrero's book.