A counterexample for two conjectures about stability
James Weldon Demmel · IEEE Transactions on Automatic Control · 1987
One measure of the stability of a matrixAis the distance to the nearest unstable matrixB. Recently Van Loan presented an algorithm to computeBwhich 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 toBby an arbitrary amount. The same counterexample invalidates another conjecture and algorithm of the author.