Optimizing matrix stability

Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.