=+10. From a periodic sequence ck with period n, form the circulant matrix C =

⎛

⎜⎜⎝

c0 cn−1 cn−2 ··· c1 c1 c0 cn−1 ··· c2

.

.

. .

.

. .

.

. .

.

.

cn−1 cn−2 cn−3 ··· c0

⎞

⎟⎟⎠ .

For un = e2πi/n and m satisfying 0 ≤ m ≤ n − 1, show that the vector (u0m n , u1m n ,...,u(n−1)m n )t is an eigenvector of C with eigenvalue ncˆm. From this fact deduce that the circulant matrix C can be written in the diagonal form C = UDU∗, where D is the diagonal matrix with kth diagonal entry ncˆk−1, U is the unitary matrix with entry u(j−1)(k−1) n /

√n in row j and column k, and U∗ is the conjugate transpose of U.