Suppose we have diagonal square($n \times n$) matrix $A$. (entries on diagonal are non-zero.)
1) Can there be non-diagonal square matrix $C$ that there exists non-zero vector $B$ with all entries non-negative of size ($1 \times n$) that satisfies $BC = BA$? And for any $A$, does $C$ always exist?
2) Continuing from 1), is there any such $C$ that $BC=BA$ all the time regardless of the choice of $B$? (Again $B$ has all entries non-negative and is not zero vector.) Again, for any $A$, does $C$ always exist?
Edit: Of course the case would be divided into two: when entries of $A$ on diagonal are same and the other. I would like to consider both cases.