Suppose $A,B\in SL(3,F\_q)$, where $F\_q$ is the finite field of order $q$ and $SL(3,F\_q)$, the group of matrices with determinant one and entries from $F\_q$ , are such that $A$ has eigenvalues in $F\_q$ and $B$ has eigenvalues in $\overline{F\_q}\setminus F\_q$. Also, $A$ is diagonalizable over $F\_q$ and $B$ is diagonalizable over $\overline{F\_q}$, the closure of $F\_q$. I am trying to show that $A$ and $B$ are not simultaneously diagonalizable by a matrix $P\in SL(3,\overline{F\_q})$ (i.e., $\nexists P\in SL(3,\overline{F\_q})$ such that $(PAP^{-1},PBP^{-1})$ are diagonal). I am considering the approach of looking at the $F\_q$ algebra generated by $A$ and $B$ and trying to show it is not isomorphic to the algebra generated by $F\_q[PAP^{-1},PBP^{-1}]$. I am looking for some reference which might be helpful in proving the above. Most of the results I saw had been about irreducible representations which is not helpful for my case. I would appreciate it if you could suggest some reference that could be helpful.
Thanks in advance for your time.



Simultaneous similarity of matrices over finite fields
I think there is a simpler way to see what is going on: first of all, the hypotheses force the characteristic polynomial of $B$ to be irreducible of degree $3$ over $F\_{q}.$ On the other hand, if $PAP^{-1}$ and $PBP^{-1}$ are both diagonal, then $PAP^{-1}$ and $PBP^{-1}$ certainly commute. Hence $A$ and $B$ already commute. Since $B$ has irreducible characteristic polynomial, (which is also its minimum polynomial in this situation), and since $B$ leaves every eigenspace of $A$ invariant (as $A$ and $B$ commute), this forces (given the hypotheses) $A$ to have the form $\lambda I$ for some $\lambda \in F\_{q}$.


