Maximum value of a parameter in a cyclic inequality?

Question

I want to figure out the set of possible values for $\alpha$ so that the inequality \begin{align*} \sum_{i=1}^{n}x_i^2 + 2\alpha\sum_{i=1}^{n-1}x_{i}x_{i+1} \ge 0 \end{align*}

Answer 1

The inequality is written as $x^\mathsf{T} A x \ge 0, \ x\in \mathbb{R}^n$ where $A$ is a symmetric tridiagonal Toeplitz matrix. (Note: The diagonal entries of $A$ are all $1$. The subdiagonal and superdiagonal entries of $A$ are all $\alpha$.)
See: https://en.wikipedia.org/wiki/Tridiagonal_matrix

All eigenvalues of $A$ is $$1 - 2\alpha \cos \frac{k\pi}{n+1}, \ k=1, 2, \cdots, n.$$

It is easy to prove that $x^\mathsf{T} A x \ge 0, \ x\in \mathbb{R}^n$ if and only if $$- \frac{1}{2\cos \frac{\pi}{n+1}} \le \alpha \le \frac{1}{2\cos \frac{\pi}{n+1}}.$$