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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.
7
votes
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 …