Consider a matrix A with N orthonormal eigenvectors {xi} and eigenvalues {λi}. Construct the matrix S from the eigenvectors of A as S = [x1    x2    · · ·    xN], where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S−1 exists. Now consider the action of An on a general vector y in this N dimensional space. We wish to evaluate Any. It is proposed that we start by expanding y as provided (image). Explain why y can be expanded as shown (quote relevant theorems/results). Find an expression for the coefficients ai in terms of a suitable inner product.

Consider a matrix A with N orthonormal eigenvectors {xⁱ} and eigenvalues {λⁱ}. Construct the matrix S from the eigenvectors of A as S = [x¹    x²    · · ·    x^N], where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S⁻¹ exists.

Now consider the action of A_n on a general vector y in this N dimensional space. We wish to evaluate Aⁿy. It is proposed that we start by expanding y as provided (image).

Explain why y can be expanded as shown (quote relevant theorems/results). Find an expression for the coefficients a_i in terms of a suitable inner product.