Proving similarity of $A$ and $A^T$

Question

$A \in \mathbb{C}^{n,n}$ is upper triangular and all it's elements on main diagonal are different. I have to prove that $A$ and $A^T$ are similar. I know that $A$ and $A^T$ have same characteristic polynomial but I don't know how to prove what I have to prove. Any hints?

Answer 1

Let $a_1,\ldots,a_n$ be the entries of the main diagonal of $A$. Since they are distinct, $A$ is similar to the diagonal matrix whose entries of the main diagonal are $a_1,\ldots,a_n$. And the same thing applies to $A^T$. Therefore, $A$ and $A^T$ are similar.