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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 $f$.

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possible duplicate of Eigenvalues of an interesting real symmetric matrix –  Davide Giraudo Dec 5 '12 at 16:36
Hi, Timothy. You may see here to learn how to make the formatting or typesetting of your question nicer. For your current question, formatting it is not that difficult. All you need is to enclose your math symbols by pairs of dollar signs. (BTW, your question was downvoted by someone while I was typing this comment. I wasn't the one who cast the downvote.) –  user1551 Dec 5 '12 at 16:37
This sounds like you are assigning us a problem with no motivation and no work that you have tried and none of your thoughts in general on the problem. Perhaps you could give us more information on what you have done? –  Jebruho Dec 5 '12 at 16:41
I have calculated f(e_1) , f(e_2).....,f(e_n) and getting the corresponding matrix , but I am unable to calculate the eigenvalues of this matrix –  Ester Dec 5 '12 at 17:42
@DavideGiraudo The cyclic conditions $x_{n+1}=x_1$ and $x_0=x_n$ make this a different question, no? –  Did Dec 5 '12 at 18:58

1 Answer 1

up vote 1 down vote accepted

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 $$ \mathrm{Spectrum}(f)=\{-4\sin^2(k\pi/n)\mid 0\leqslant k\leqslant n-1\}. $$

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