assume we have $n^2$ matrixs $A_{ij},1\le i,j\le n$ each one is not zero matrix and the size is $n\times n$ . they satisfy the following condition:
$A_{ij}A_{pq}=\delta_{jp}A_{iq}$
where $\delta_{st}=1$ if $s=t$ otherwise $\delta_{st}=0$
prove that there is a reversible matrix $P$ satisfies that $P^{-1}A_{ij}P=E_{ij}$ where $E_{ij}$ donates the matrix that's all zero except the $ij$ entry which is 1.
here is what i try so far:
i figured out since $A_{ii}$ and $\sum A_{ii}$ are both idempotent matrixs we have $\sum rankA_{ii}=rank \sum A_{ii}$，this show each $A_{ii}$ rank one and since $A_{ii}A_{jj}=A_{jj}A_{ii}=0$ they can simultaneous diagonalization . then we get a reversible matrix Q satisfies $Q^{-1}A_{ii}Q=E_{ii}$. consider $A_{ii}A_{ij}A_{jj}=A_{ij}$ we get $A_{ij}=a_{ij}E_{ij},i\not=j,a_{ij}\not=0$ . now i figured out $a_{ij}$ may not be 1. so we must multiply some reversible matrix on Q and i stuck here.
i wonder how to do next or if there is other way to solve , thanks!