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I am trying to find the eigenvalues/eigenvectors of a NxN tridiagonal symmetric "nearly" Toeplitz matrix, except that a modification on the top left corner \begin{pmatrix} a^2 & -a \\ -a & 1+a^2 & -a \\ & -a & \ddots & \ddots \\ & & \ddots & \ddots & -a \\ & & & -a & 1+a^2 \end{pmatrix} and $|a|<1$. For the case of tridiagonal symmetric Toeplitz matrix, we have analytical expression of both eigenvalues and eigenvectors. With some rank-1 perturbation on the top-left corner, maybe the best we could do is to find numerical solution of eigenvalues/eigenvectors. From numerical experience, 1 out of N eigenvalues will fall out of range, accordingly, the associated eigenvector pair behave unusual (should be sinusoidal functions without perturbation). I hope someone could shed some light on this problem, is there any to determine this specific least eigenvalue/eigenvector? Or, could we show that this least eigenvalue decays as N gets bigger? I found some similar problem here: Eigenvectors of almost-Toeplitz tridiagonal matrix., however, we don't have anything analytical.

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