Are $A$ and $A^\top$ similar?

Question

Let $K$ be a field and $A$ a square matrix with entries in $K$. Then A and $A^\top$ have the same characteristic polynomial. What do we know about similarity? Do you have an example where $A$ and $A^\top$ are not similar?

Answer 1

$A$ and $A^T$ are necessarily similar. To show this, it suffices to note that $$ \dim \ker [(A - \lambda I)^k] = \dim \ker [(A^T - \lambda I)^k] $$ for all $\lambda$ taken from the algebraic closure of $K$ and all $k \in \Bbb N$.