Diagonalizable matrix over R

Question

I had study about solving linear difference equation system by using results of linear algebra. But I have a problem with the following exercise:

Suppose that $A$ is a matrix with real entries, $A$ is diagonalizable over $\mathbb{C}$ and its characteric polynomial is separable over $\mathbb{R}$. Prove that $A$ is diagonalizable over $\mathbb{R}$.

Please give me some hint!

Answer 1

There are $P,Q\in M_n(\mathbb{R})$ s.t. $P+iQ\in GL_n(\mathbb{C})$ and $(P+iQ)^{-1}A(P+iQ)=diag((\lambda_i))$, where $\lambda_i\in\mathbb{R}$ (why?).

Show that, for every $u\in\mathbb{R}$, $A(P+uQ)=(P+uQ)D$.

Show that there is $u\in \mathbb{R}$ s.t. $P+uQ\in GL_n(\mathbb{R})$.

Conclude.