Suppose the matrix A is used to transform points in the plane iteratively. That is, given a point v, consider the sequence vn = Anv. Letting U = [u1 u2] so that ui is an eigenvector associated to λi and letting v = c1u1 + c2u2 what is a simple expressions for an and bn so that vn = Anv = anu1 + bnu2. If A and B are similar square matrices and say S witnesses this, that is A = SBS−1 , show that if (λ, v) is an eigenvalue/eigenvector pair for A, then (λ, S−1v) is an eigenvalue/eigenvector pair for B.

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