Let V be a finite dimensional vector space over C. Let D, N e L(V). Suppose that the operator D is diagonalizable, the operator N is nilpotent, and DN = ND. Let 11,..., Am be the distinct eigenvalues...

Extracted text: Let V be a finite dimensional vector space over C. Let D, N e L(V). Suppose that the operator D is diagonalizable, the operator N is nilpotent, and DN = ND. Let 11,..., Am be the distinct eigenvalues of D. Define T e L(V) by T = D+N. (a) Prove that E(^¡,D) is invariant under N for every j. (b) Prove first that E(A¡, D) C G(Aj, T), and then deduce that E(A;, D) = G(^¡, T) for every j.