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Consider the Laplacian matrix of the path graph: $$ L = \begin{bmatrix} 1 & -1 & 0 & \cdots & 0 & 0\\ -1 & 2 & -1 & \cdots & 0 & 0\\ 0 & -1 & 2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots &\ddots&\vdots&\vdots\\ 0 & 0 & 0 & \cdots & 2 & -1\\ 0 & 0 & 0 & \cdots & -1 & 1 \end{bmatrix} $$

I want to find the eigendecomposition of this matrix. As far as I know, there are no general formulas for bisymmetric tridiagonal matrices. It's almost Toeplitz and I know the formula for tridiagonal Toeplitz matrices.

Any resources or pointers for this?

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    $\begingroup$ Can you confirm that the signs of the matrix entries are correct? $\endgroup$
    – Bertoldo Baccalà
    Commented Sep 25 at 17:50
  • $\begingroup$ The off diagonals were flipped, thank you for pointing it out. $\endgroup$
    – user123
    Commented Sep 25 at 17:54
  • $\begingroup$ And can you confirm the entries top left and bottom right are really 1? $\endgroup$
    – Bertoldo Baccalà
    Commented Sep 25 at 18:10
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    $\begingroup$ Yes, this is the difficulty. The endpoints of the path have degree 1. $\endgroup$
    – user123
    Commented Sep 25 at 18:13
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    $\begingroup$ This is the same as a StackExchange question that points to a complete solution by Jiang. $\endgroup$
    – Nathaniel Johnston
    Commented Sep 25 at 18:18

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