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I looking to find the eigenvalues of a matrix of the following form:

$$\begin{pmatrix} a & b & 0 & 0&0 \\ b &a &b &0 &0\\0&b&a &b&0\\0&0&b&a&b\\ 0&0&0&b&a\end{pmatrix}$$ I am looking to do so in the case of a matrix of this form of size $N\times N$. Is there any way to derive these eigenvalues in the general $N$ case?

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Here's a paper that discusses that topic in great generality:

https://cpsc.yale.edu/sites/default/files/files/tr1460.pdf

and a wikipedia article: https://en.wikipedia.org/wiki/Tridiagonal_matrix#cite_note-7

Check out the section on eigenvalues, noting that your matrix has three important properties:

  1. Tridiagonal
  2. Symmetric
  3. Toeplitz

That means that the eigenvalues are

$$ \lambda_i = a - 2b \cos\left(\frac{k\pi}{n+1}\right) \text{ for } k = 1, \dots, n. $$

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