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6. Let A be an n x n matrix with distinct real eigen- values A₁, A2,...,. Let λ be a scalar that is not an eigenvalue of
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6. Let A be an n x n matrix with distinct real eigen- values A₁, A2,...,. Let λ be a scalar that is not an eigenvalue of
6. Let A be an n x n matrix with distinct real eigen- values A₁, A2,...,. Let λ be a scalar that is not an eigenvalue of A and let B = (A-AI). Show that (a) the scalars = 1/(λ), j = 1,...,n are the eigenvalues of B. (b) if x, is an eigenvector of B belonging to ,, then x is an eigenvector of A belonging to A. (c) if the power method is applied to B, then the sequence of vectors will converge to an eigen- vector of A belonging to the eigenvalue that is closest to A. [The convergence will be rapid if is much closer to one λ than to any of the oth- ers. This method of computing eigenvectors by using powers of (A-AI) is called the inverse power method.]