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?

  1. Suppose that $A$ has a cyclic vector, a vector $x$ such that the vectors $x,Ax,\ldots,Ax^{n-1}$ are a basis. Show that the matrix representing the action of $A$ relative to this basis is real. (Look up companion matrices if you get stuck here.)

  2. Look up Frobenius normal form (or rational normal form).

  • $\begingroup$ I solved it by showing it is similiar to the rational form, and it belongs to $\mathbb{R}$ by every characteristic value is real. Thank you very much Chris Godsil $\endgroup$ – Block Jeong Nov 24 '13 at 4:53

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