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.

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A: Consider a matrix A with N orthonormal eigenvectors {xi} and eigenvalues {λi}. Construct the matrix…

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