Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose.
I spent quite a bit of time on it now, and still could not prove it. At the risk of devaluing myself, might I ask someone else to show me a proof?
This question has a nice answer using the theory of modules over a PID. Clearly the Smith normal forms (over K[X]) of XIn−A and of XIn−AT are the same (by symmetry). Therefore A and AT have the same invariant factors, thus the same rational canonical form*, and hence they are similar over K.
*The Wikipedia article at the link badly needs rewriting.