Show that $A$ is similar to $A^T$

Question

Assume $A$ is diagonalizable. Show that $A$ is similar to $A^T$

Any help would be very appreciated. I know for a matrix to be diagonalizable it has to be a square matrix so the # of rows and columns stay the same for $A^T$.

Answer 1

Hint: $$A=PDP^{-1}$$ $$A^T=(P^{-1})^TDP^T$$ Express $D$ in terms of $A^T$ and substitute to the first equation.

Note that $\left(P^{-1}\right)^{T}=\left(P^T\right)^{-1}.$ It is also commonly denoted by $P^{-T}$.

Answer 2

Given that $A$ is diagonalizable, so there exists an invertible matrix $P$ such that $$D=P^{-1}AP$$ where $D$ is a diagonal matrix.
So we obtain $$A=PDP^{-1}$$ Then $$A^T=(PDP^{-1})^T=(P^{-1})^TD^TP^T=(P^{-1})^TDP^T=(P^{-1})^T(P^{-1}AP)P^T=Q^{-1}AQ$$ where $Q=PP^T$ is an invertible matrix.
So by definition, $A$ is similar to $A^T$.