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.
Solution
Let \[ R=\operatorname{diag}(C(p_1),\ldots,C(p_r)) \] be the rational canonical form of \(A\) over \(\mathbb C\), with invariant factors \(p_i\in\mathbb C[x]\) [@HK71].
1. Every invariant factor has real coefficients.
Each invariant factor \(p_i\) divides the characteristic polynomial of \(A\). Over \(\mathbb C\), it therefore factors as \[ p_i(x)=\prod_j(x-\lambda_j)^{m_j} \] for characteristic values \(\lambda_j\) of \(A\). By hypothesis every \(\lambda_j\) is real. Since \(p_i\) is monic, the displayed product has real coefficients. Hence \[ p_i\in\mathbb R[x] \] for every \(i\).
2. The rational form is a real matrix.
The companion matrix of a monic polynomial has entries consisting only of \(0\), \(1\), and the negatives of its coefficients. Since every \(p_i\) has real coefficients, every block \(C(p_i)\) has real entries. Therefore \[ R\in M_n(\mathbb R). \] By definition of rational form there is some \(P\in GL_n(\mathbb C)\) such that \[ P^{-1}AP=R. \] Thus \(A\) is similar over \(\mathbb C\) to the real matrix \(R\), proving the claim.