Let and . Show that these matrices have the same characteristic polynomial, but that one is diagonalizable and the other not. Compute the Jordan canonical form for each.
Let and . Evidently we have and , which are both in Jordan canonical form.
Recall that characteristic polynomials are invariant under conjugation; in particular, evidently both and have characteristic polynomial . By Corollary 24 in D&F, is not diagonalizable.