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Let $A$, $B$ and $C$ be complex matrices such that $C\neq 0, $ $AC=CB$. Prove that $A$ and $B$ have a common eigenvalue.
There is a hint in the question, these facts can be used for the prove:
- For a complex matrices $A, B$, If $AB = 0$, and $B$ is invertible, $A = 0$.
- For a complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k $.