A counterexample for two conjectures about stability

One measure of the stability of a matrix A is the distance to the nearest unstable matrix B . Recently Van Loan presented an algorithm to compute B which depended on a conjecture about the location of its eigenvalues. We provide a counterexample to this conjecture which shows that the algorithm may overestimate the distance to B by an arbitrary amount. The same counterexample invalidates another conjecture and algorithm of the author.