The square matrix with entries given by
(1)
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for , 1, 2, ..., , where i is the imaginary number , and normalized by to make it a unitary. The Fourier matrix is given by
(2)
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and the matrix by
(3)
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(4)
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In general,
(5)
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with
(6)
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where is the identity matrix and is the diagonal matrix with entries 1, , ..., . Note that the factorization (which is the basis of the fast Fourier transform) has two copies of in the center factor matrix.