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Given a tridiagonal $(n-1)\times (n-1)$ matrix

$A=n^2\begin{bmatrix} -2&1&0&0&0&\dots&0 \\ 1&-2&1&0&0&\dots& 0\\ 0&1&-2&1&0&\dots& 0\\ 0&0&1&-2&1&\dots& 0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots \\ 0&0&0&0&\dots&1&-2\end{bmatrix}$

its eigenvalues are given as $\lambda_k=-4\sin^2(\frac{\pi k}{2n}),\, k=1,\dots,n-1.$

The question is to find the spectral radius for a Gauss-Siedel matrix, $G=(D-E)^{-1}F$. Here $D$ is diagonal matrix (diagonal of $A$), $E$ is lower-diagonal part of $A$ and $F$ is upper-diagonal part.

So I can't figure out how to use the eigenvalues here. Clearly, we need to find a way to evaluate the eigenvalues of $G$. Are there any properties that could be applied here?

Ilia
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    Minor correction: there is no way for you to have $n$ eigenvalues with an $(n-1) \times (n-1)$ matrix... – Ian Dec 11 '16 at 02:41
  • @Ian you're absolutely correct, my mistake – Ilia Dec 11 '16 at 02:42
  • A much less important correction; you should write $\lambda_k$. :) Anyway, I think you may be able to write down what $(D-E)^{-1}$ is explicitly. – Ian Dec 11 '16 at 02:44

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