# Linear transformations that are diagonalizable and nilpotent

Let $V$ be a vector space of finite dimension. Find all linear transformations $T:V\rightarrow V$ that are both diagonalizable and nilpotent.

I was thinking $T=0$. But are there other such transformations?

• What does a diagonal nilpotent matrix have to be? – Rankeya Apr 30 '15 at 19:32

If $T$ is diagonalizable, then with respect to some basis of $V$, it look like diag($a_1,...,a_n$). Then $T^k$ looks like diag($a_1^k,...,a_n^k$). So $T^k = 0$ implies that each $a_r^k = 0$. Since we're over a field, $T = 0$.
Hint: If $T$ is nilpotent, then its only eigenvalue can be $0$.
Now, suppose that $T$ has a basis of eigenvectors.