Let nilpotent linear transformation over some field, let assume that .
Prove that nilpotent if and only if the characteristic polynomial of is
Thank you!
Remember, T is nilpotent if there exists some such that . Use this along with the Cayley-Hamilton theorem to prove this fact (the <= side should be trivial).
Remember, T is nilpotent if there exists some such that . Use this along with the Cayley-Hamilton theorem to prove this fact (the <= side should be trivial).
If is the char. polynomial of T then by Cayley-Hamilton, therefore T is nilpotent by definition.
For the other direction, simply prove that T can not have any non-zero eigenvalues (use the fact that if is an eigenvalue of T then is an eigenvalue of ).