The row and column indices in the nxn Fourier matrix A run from 0 to n-1, and the i,j entry is E^ij, where E^ij = e^(2*PI*i/n). This matrix solves the following interpolation problem: Given complex numbers b_0, ... b_(n-1), find a complex polynomial f(t) = c_0 + c_1 + ... + c_(n-1) t^(n-1) such that f(E^v) = b_v.
(i) Explain how the matrix solves the problem.
(ii) Prove that A is symmetric and normal, and compute A^2.
(iii) Determine the eigenvalues of A.
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Fourier matrix is contemplated in this solution.