Proof of a matrix and it's transpose being similar. [duplicate]

Prove or disprove: If $A$ is an $n\times n$, then $A$ and $A^T$ are similar.

Definition of Two Similar Matrices: Let $A$ and $B$ be two $n\times n$ matrices. Then $A$ and $B$ are similar if there exists a nonsingular matrix $S$, such that $A=S^{-1}BS$.

I feel like there aren't similar. But I am not sure. Can I get some help?

Edit: So I have been googling an answer. And I see that most theorems state that this claim is true if the entries of $A$ are complex. Is it true if they were? i.e. Is $A$ and $A^T$ similar, when $A$ has real components?

Solution 1:

They are always similar, the proof is given in this article: ON THE SIMILARITY TRANSFORMATION BETWEEN A MATRIX AND ITS TRANSPOSE, OLGA TAUSSKY AND HANS ZASSENHAUS, Caltech, 1959.