Higham there adds the condition that X, Y be well-conditioned (as they are in the SVD, for example). That's a problem with the Jordan decomposition A = X J X^-1

Higham there adds the condition that X, Y be well-conditioned (as they are in the SVD, for example). That's a problem with the Jordan decomposition A = X J X^-1: the X is non-singular in theory, but can be ill-conditioned, and thus the Jordan decomposition is not stable and not really used in practice (as highlighted in the original article).