Question: Define the shift map S: Cn Cn by S(v1, v2, ..., vn-1, vn)T = (v2, v3, ..., vn, v1)T. (a) Prove that S is

Define the shift map S: Cn → Cn by
S(v1, v2, ..., vn-1, vn)T = (v2, v3, ..., vn, v1)T.
(a) Prove that S is a linear map, and write down its matrix representation A.
(b) Prove that A is an orthogonal matrix.
(c) Prove that the sampled exponential vectors ω0, ...., ωn-1 in (5.90) form an eigenvector basis of A. What are the eigenvalues?

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