Let be an matrix over a field . Suppose is nilpotent; say for some natural number . Prove that is similar to a matrix whose first superdiagonal consists of 0 and 1 and whose remaining entries are 0.
Suppose is an eigenvalue of . By this previous exercise, is an eigenvalue of , and thus , so that . That is, all of the eigenvalues of are 0.
Thus, if is the Jordan canonical form of , then the Jordan blocks of have 1 on the first superdiagonal and 0 elsewhere.