The difference map ∆: Cn → Cn is defined as ∆ = S - I, where S is the shift map of Exercise 8.2.13.
(a) Write down the matrix corresponding to A.
(b) Prove that the sampled exponential vectors w0, ...,wn-1 from (5.90) form an eigenvector basis of ∆. What are the eigenvalues?
(c) Prove that K = ∆T ∆ has the same eigenvectors as ∆. What are its eigenvalues? Is K positive definite?
(d) According to Theorem 8.20 the eigenvectors of a symmetric matrix are real and orthogonal. Use this to explain the orthogonality of the sampled exponential vectors. But, why aren't they real?