# Exercise 13, Section 7.2 of Hoffman’s Linear Algebra

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.

My attempt: Suppose $$A\in M_n(\Bbb{C})$$. Let $$f$$ and $$m$$ be characteristic and minimal polynomial of $$A$$. Since $$\Bbb{C}$$ is algebraically closed and $$f\in \Bbb{C}[x]$$, we have $$f$$ splits into linear factors. That is $$f=b(x-c_1)^{d_1}\cdots (x-c_k)^{d_k}$$, where $$b,c_i\in \Bbb{C}$$ and $$d_i\geq 1$$. Since $$f$$ is a monic polynomial, $$b=1$$. So $$c_1,…,c_k$$ are eigenvalues of $$A$$. By hypothesis, $$c_i\in \Bbb{R}$$, for all $$i\in J_k$$. Since $$f$$ and $$m$$ have same roots, we have $$m=(x-c_1)^{e_1}\cdots (x-c_k)^{e_k}$$, where $$e_i\geq 1$$. So $$m\in \Bbb{R}[x]$$. Define $$T:\Bbb{C}^n\to \Bbb{C}^n$$ such that $$[T]_B=A$$, where $$B$$ is canonical basis of $$\Bbb{C}^n$$. By theorem 3 section 7.2, $$\exists \alpha_1,…,\alpha_r\in \Bbb{C}^n\setminus \{0\}$$ with respective $$T$$-annihilators $$p_1,…,p_r$$ such that $$V=Z(\alpha_1;T)\oplus \cdots Z(\alpha_r;T)$$, and $$p_{i+1}|p_i$$, and $$p_1=m$$. Since $$p_i|p_1=m$$, we have $$p_i=(x-c_1)^{g_1}\cdots (x-c_k)^{g_k}$$, where $$0\leq g_i\leq e_i$$. So $$p_i\in \Bbb{R}[x]$$, $$\forall i\in J_r$$. Then $$\exists \mathcal{B}$$ basis of $$\Bbb{C}^n$$ such that $$[T]_{\mathcal{B}}=\begin{bmatrix}A_1& & \\ & \ddots & \\ & & A_r \\ \end{bmatrix}$$ where $$A_i$$ is $$k_i \times k_i$$ companion matrix of $$p_i$$. So $$A_i\in M_{k_i\times k_i}(\Bbb{R})$$, $$\forall i\in J_r$$. Thus $$[T]_\mathcal{B}\in M_n(\Bbb{R})$$. Hence $$A=[T]_B$$ is similar over $$\Bbb{C}$$ to $$[T]_\mathcal{B}$$. Is my proof correct?

It is correct but could be shorter: by theorem 5 of the same section, $$A$$ is similar over $$\Bbb C$$ to a matrix $$\begin{bmatrix}A_1& & \\ & \ddots & \\ & & A_r \\ \end{bmatrix}$$ where each $$A_i$$ is the companion matrix of some monic polynomial $$p_i.$$ Since the roots of the characteristic polynomial $$\prod p_i$$ are real, so are the coefficients of each $$p_i$$ hence the entries of $$A_i.$$
• Thank you so much for the answer. Your proof is so slick! I think we need to use “if $A_i$ is companion matrix of monic polynomial $p_i$, then $p_i$ is characteristic polynomial of $A_i$” result to show characteristic polynomial of $A$ is $\prod_{i=1}^rp_i$. And “roots of characteristic polynomial $\prod p_i$ are real$\implies$coefficients of each $p_i$ are real” uses splits property of polynomial over algebraically closed field. Let me know if I’m correct? Commented Mar 4, 2023 at 18:35
• I agree we need to use "the characteristic of the companion matrix of a monic polynomial $p$ is $p$". As for the second sentence, I would rather say that when talking about the roots of $p_i$ (and saying they are real), we already know $p_i$ is split over $\Bbb C.$ Commented Mar 4, 2023 at 18:42
• Yeah. I just wanted to make sure that splitting of $p_i$ is assumed implicitly. Commented Mar 4, 2023 at 18:46