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.
Categories: math.CA math.FA math.OA