I have found an exercice in a calculus book, which I have problems to solve.
$\text{Let}\, R:\mathbb{R}^n-{0}\to \mathbb{R},\quad R(x)=\frac{x^tAx}{x^tx} = \frac{\langle x,Ax\rangle}{\langle x,x\rangle},$ where $\langle \cdot,\cdot \rangle$ is the euclidean inner product, $A\in\mathbb{R}^{n\times n}$.
$1)$ $R$ has a minimum in $\mathbb{R}-{0}$.
$2)$ Every critical point of $R$ is an Eigenvector of $\frac{1}{2}(A^t+A)$ corresponding to an Eigenvalue of $A$. In particular, every symmetric real matrix has real Eigenvalues.
I have solved $1)$ (by seeing that it is sufficient to study $R|S^n$ and $S^n$ is compact, so $R$ has a minimum and maximum which it attains.) For $2)$ I can't find the desired result. I know that $\frac{1}{2}(A^t+A)$ is symmetric, and that $\langle x,Ax \rangle=\langle x,A^tx \rangle$.