Question: Let V be a finite dimensional vector space over C. Let D, N L(V). Suppose the operator D is diagonalizable, and operator N is nilpotent.

Let V be a finite dimensional vector space over C. Let D, N L(V). Suppose the operator D is diagonalizable, and operator N is nilpotent. Also DN = ND. We are going to let 1,...,m be the distinct eigenvalues of D. We define ZL(V) by Z=D+N.

(a) Prove that E(j, D) is invariant under N for every j.

(b) Prove first that E(j, D) G(j, Z), and then deduce that E(j, D) = G(j, Z) for every j.

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