Proving that matrix is similar to its transpose

Question

One way to prove that a matrix and its transpose are similar is to apply polynomial matrix theory.

I wonder if there is another way to do that. Thanks.

Answer 1

The standard way to do it is to prove that $A$ and $A^t$ have the same Jordan canonical form. In turn, you can prove this showing that for all $\lambda$ and for all $n$

$(A-\lambda I)^n $ and $ (A^T-\lambda I)^n$ have the same rank.