Spectrum of a Special Matrix

Question

Let $f : \mathbb{C}^n \longrightarrow \mathbb{C}^n$ be defined by $\left(fx\right)_m = x_{m+1} - 2x_m + x_{m-1},\;m \in \mathbb{N}$ where $x=(x_1,\,\dots\,,x_n)^T,\;x_{n+1}=x_1,\;x_0 = x_n$ .

Find the spectrum of $f$.

Answer 1

Call $A$ the set of $n$th roots of unity and, for every $a$ in $A$, $x_a$ the vector of coordinates $(x_a)_k=a^k$ for $1\leqslant k\leqslant n$. Then $fx_a=\lambda_ax_a$ with $\lambda_a=a+a^{-1}-2$. The family $(x_a)_{a\in A}$ is linearly independent hence the spectrum of $f$ is the collection $(\lambda_a)_{a\in A}$.

Since $\mathrm e^{\mathrm i\theta}+\mathrm e^{-\mathrm i\theta}-2=-4\sin^2(\theta/2)$ for every $\theta$, one gets $$ \mathrm{Spectrum}(f)=\{-4\sin^2(k\pi/n)\mid 0\leqslant k\leqslant n-1\}. $$