A famous result in linear algebra is the following.

An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.

I know one proof using the Smith Normal Form (SNF). However, I want to find an elementary proof avoiding any concepts related to the SNF. My question is: is there an elementary way to prove this?

The requirements are:

• Do NOT use the structure theorem over PID.

• Do NOT use the Smith Normal form (nor Jordan canonical form).

• Do NOT use the concept of invariant factors.

• Provide an explicit invertible matrix $P$ such that $A=PA^T P^{-1}$.