Let $A$ be an $n \times n$ matrix with complex entries. Prove that if every characteristic value of $A$ is real, then $A$ is similar to a matrix with real entries.
This problem is from Sec.7.2 "Cyclic decompsition and the rational form" of Linear Algebra, Kunze and Hoffman. I can't even get a clue how to approach to this problem. Any body could give any hint?