  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.
