Abstract Algebra Question

Let N : Cn - C" be a nilpotent operator. (Recall that N is nilpo- tent if N" = 0 for some integer r E N). (a) Show that the number of Jordan blocks in the canonical Jordan form of N is equal to dim ker (N). (b) Show that the size of the largest Jordan block of N is equal to the nilpotency index of N (Recall that the nilpotency index of N is the smallest r E N such that N" = 0). (c) Find the Jordan canonical form of the nilpotent operator N : C6 _ C' given by the matrix OOOON 2 1 - O LOLHON LO OOOO. LOWIN