Special Properties of Real Matrices With Real Distinct Eigenvalues Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do with the realness of the eigenvalues?
In short, I'm searching for the properties that comes from the realness of the eigenvalues. Can you at least suggest me a reference for the topic?
Note: This is a little different from the question What do real eigenvalues imply for a matrix as MorganRodgers pointed out in the comments. My question requires that the eigenvalues be distinct and the answer relies on that.
 A: Let $A\in M_n(\mathbb{R})$. 
$A$ has $n$ distinct real eigenvalues iff 
there are $\lambda\in\mathbb{R}$ and $P,Q$ real symmetric $>0$ s.t. $(A+\lambda I)^2=PQ$ AND the dimension of the commutant of $A$ is $n$.
EDIT 1. Proof: $(\Rightarrow)$ Clearly $A$ is diagonalizable and the dimension of its commutant is $n$. Let $\lambda\in\mathbb{R}$ s.t. $A+\lambda I$ is invertible. Then $(A+\lambda I)^2$ is diagonalizable and has positive eigenvalues. Then it can be  put in the form $PQ$.
$(\Leftarrow)$ $(A+\lambda I)^2=PQ$ implies that $(A+\lambda I)^2$ is invertible, diagonalizable and has positive eigenvalues. Then $A+\lambda I$ is invertible, diagonalizable and has real eigenvalues. Consequently $A$ is diagonalizable and has real eigenvalues; since its commutant has dimension $n$, its eigenvalues are simple.
EDIT 2. We can do better. $A$ has $n$ distinct real eigenvalues iff 
there are real symmetric $P,Q$ where $P>0$ s.t. $A=PQ$ AND the dimension of the commutant of $A$ is $n$.
Proof. $(\Rightarrow)$ $A=RDR^{-1}$, where $R,D$ are real and $D$ is diagonal, implies $A=PQ$ with $P=RR^T,Q=R^{-T}DR^{-1}$. (This proof is due to O. Taussky). 
$(\Leftarrow)$ is easy.
