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

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 $n\ge 2$, 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.