The Distance from a Rank Projection to the Nilpotent Operators on

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 $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$ . 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.