Optimizing matrix stability

Given an affine subspace of square matrices, we consider the problem of min imizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a sing le 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 s pace of all matrices, they appear naturally in spectral abscissa minimizati on.