Variation in Jordan structure under congruence: The Nilpotent case Original Research Article

For each congruence class containing a nilpotent matrix, all possible nilpotent Jordan structures occurring in that class are determined. The only two constraints are: a fixed number of Jordan blocks, determined by the rank of the class, and a minimum number of one-by-one blocks, determined by the dimension of the intersection of the null spaces of A and A*, for A representative of the class. An important tool, that may be of independent interest, is a unitary similarity version of the Jordan canonical form.