# The Distance from a Rank $n-1$ Projection to the Nilpotent Operators on $\mathbb{C}^n$

## TL;DR

This paper determines the exact distance from rank n-1 projections to nilpotent matrices in complex n-dimensional space, extending previous results for rank-one projections and exploring unitary equivalence of minimal pairs.

## Contribution

It generalizes MacDonald's formula to higher rank projections and characterizes minimal distance pairs as unitarily equivalent, with discussions on intermediate ranks.

## Key findings

- Distance from rank n-1 projection to nilpotents is  sec((n/(n-1)+2))
- Constructed minimal distance pairs are unitarily equivalent
- Results suggest possible extensions to intermediate rank projections

## Abstract

Building on MacDonald's formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$ is $\frac{1}{2}\sec\left(\frac{\pi}{\frac{n}{n-1}+2}\right)$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where $Q$ is a projection of rank $n-1$ and $T\in\mathbb{M}_n(\mathbb{C})$ is a nilpotent of minimal distance to $Q$. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.