Let $A,B,C \ne 0 \in M_{n} (\mathbb C)$ and $g(X)\in \mathbb C[X]$ such that $AC=CB$
I need to prove that for every $j=1,2,3..$ the matrices implies $A^jC=CB^j$ and $g(A)C=Cg(B)$ and prove that A and B have a common eigenvalue.
If $AC=CB$ does it mean that $C$ must be diagonal matrix or at list symmetric?
I tried to use Jordan form to solve it, but I didn't succeed.
Thanks again.