The linkage of symmetric matrix and its eigenvalues

I want to know if the following claim is correct ( I believe that no):

Let A be a symmetric $n \times n$ matrix , $x \in \mathbb{R}^{n}$. The eigenvalues of $A$ are real but does it mean that:

$$ (A x, x) \geqslant \lambda_{\min }\|x\|^{2} $$

$$ \lambda_{\min }=\min _{1 \leqslant i \leqslant n }\left\{\lambda_{i} ; A x_{0}=\lambda_{i} x_{i}\right\} $$

If $A$ is invertible does it change anything?

Solution 1:

An important definition is the Rayleigh Quotient of $A$:

$$R:=(Ax,x)/\|x\|^2$$

If you consider its min/max then it’s equivalent to the min/max of

$$(Ay,y)$$

over unit vectors $y$. Now expand $y$ in terms of $A$‘s eigenvectors to prove that for symmetric $A$, the Rayleigh quotient is always between the largest and smallest eigenvalues.