Let $A,B,C$ in $M_n(\mathbb C).$ Suppose $CA = BC$ and $rank(C) = r$. Show that $A$ and $B$ have at least $r$ eigenvalues in common. 
Let $A,B,C$ in $M_n(\mathbb C).$ Suppose $CA = BC$ and $rank(C) = r$.
  Show that $A$ and $B$ have at least $r$ eigenvalues (counted with
  multiplicity) in common.

I do not see how to use characteristic polynomial, and minimal polynomial does not give the number of eigenvalues counted with multiplicity. If someone has an hint...
There exist $P$ and $Q$ invertible matrices such that $C = PJ_rQ$ with $J_r$ having first $r$ diagonal coefficients equal to $1$, and $0$ elsewhere.  
N.B. : It is supposed to be able to be solved without Jordan reduction since it is not in my curriculum (anyway, it is possible it simplifies the problem).
 A: Hint: If $v$ is a $\lambda$-eigenvector of $A$, then $Cv$ (if non-zero) is a $\lambda$-eigenvector of $B$, since $BCv=CAv=\lambda Cv$. This immediately implies the statement if $A$ is diagonalizable (since a complete basis of eigenvectors of $A$
maps to an $r$-dimensional space of eigenvectors for $B$ under $C$). 
Hint 2: If $A$ is not diagonalizable, then we instead do the same tactic with generalized eigenvectors. Suppose $v$ is a generalized $\lambda$-eigenvector of $A$; that is, $v\neq 0$ but $(A-\lambda I)^n v=0$ for some positive integer $n$. Observe that $C(A−\lambda I)^n=(B−\lambda I)^nC$ for any $\lambda,n$, thus if $Cv\neq 0$, then $Cv$ is a generalized $\lambda$-eigenvector of $B$. 
A: Use elementary row/column operations to pick two invertible matrices $U$ and $V$ such that $UCV=I_r\oplus0$. Then
$$
(UCV)(V^{-1}AV)=(UBU^{-1})(UCV).\tag{1}
$$
Partition $V^{-1}AV$ and $UBU^{-1}$ into $2\times2$ block matrices so that their first sub-blocks have the same sizes as $I_r$. What does the equality $(1)$ tell you about the first block row of $V^{-1}AV$ and the first block column of $UBU^{-1}$?
