During my calculations for a proof, I want to find two permutation matrices $P_1$ and $P_2$ such that:
$$P_1 A = A P_2$$
for a given matrix $A$. What can we say about $P_1$ and $P_2$, other than it holds for $P_1=P_2=I$?
For instance, let's take $P_1$ shuffling the rows of the symmetric matrix $A$ as follows:
$$\begin{bmatrix}0 & 0 & 1\\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} a & b & c\\ b & d & e \\ c & e & f\end{bmatrix}= \begin{bmatrix} c & e & f\\ a & b & c \\ b & d & e \end{bmatrix}$$
We can see there is no way to have a $P_2$ shuffling the columns of our original symmetric matrix and have the same result. So in general it is not possible. Are there particular class of matrices $A$ or/and choice of matrices $P_1$ and $P_2$ for which the above equality holds?