Let u and v be eigenvectors of a matrix A, with corresponding eigenvalues A and and let c1 and c2 be scalars. Define

Let u and v be eigenvectors of a matrix A, with corresponding eigenvalues A λ and μ and let c1 and c2 be scalars. Define
xk = c1λku + c2μkv (k = 0, 1, 2,...)
a. What is xk+1, by definition?
b. Compute Axk from the formula for xk, and show that Axk = xk+1. This calculation will prove that the sequence {xk} defined above satisfies the difference equation xk+1 = Axk (k = 0, 1,2,...).