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A circulant matrix is a square matrix where each row has the same elements as the previous row, cyclically rotated right by one element. It is a specific kind of a square Toeplitz matrix.

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Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix

If I have read your question correctly, the second matrix is a so-called circulant matrix, and so one can read off the spectrum using known methods. Wikipedia gives you a formula that can be used. Th …
Yemon Choi's user avatar
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