In some basis $\displaystyle e_1,e_2,...,e_n$ the matrix of an operator A is an n x n lower triangular matrix with diagonal entries $\displaystyle \lambda$, subdiagonal entries 1, and 0 for all other entries. In what basis does it have Jordan canonical form?

It seems like you could just reverse the order of the basis (ie, $\displaystyle e_n,e_{n-1},...,e_1$). Is it enough to use the fact that $\displaystyle A^T$ is in the required form to prove that this is the correct basis?