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Consider the $n \times n$ Toeplitz matrix

\begin{equation} T_n = \begin{bmatrix} a & b & 0 & 0 & \cdots & 0 \\ \bar{b} & a & b & 0 & \cdots & 0\\ 0 & \bar{b} & a & b & \cdots & 0\\ \vdots &\vdots & \ddots &\ddots &\ddots &\vdots \\ 0 & 0 & 0 & 0 & \bar{b} &a \end{bmatrix}. \end{equation} I can calculate the eigenvalues which would be

\begin{equation} a + 2|b| \cos \frac{k \pi}{n+1} \quad 1\le k \le n. \end{equation} This follows from simple Fourier series calculation. However, how to calculate the eigenvectors? Is there a general formula as well?

In general what happens for eigenvectors of Hermitian Toeplitz operators? Advanced help for any help/ suggestion/ references.

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  • $\begingroup$ Eigenvalues and eigenvectors of tridiagonal Toeplitz matrices (not just the Hermitian ones) have known closed forms. See, e.g. Noschese et al. (2011), Tridiagonal Toeplitz matrices: properties and novel applications (link). $\endgroup$ – user1551 Sep 15 '17 at 6:22

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