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# Let A be an n n matrix with distinct real eigenvalues 1, 2, . . . , n. Let be a scalar ISBN: 9780136009290 436

## Solution for problem 6 Chapter 7.6

Linear Algebra with Applications | 8th Edition

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Problem 6

Let A be an n n matrix with distinct real eigenvalues 1, 2, . . . , n. Let be a scalar that is not an eigenvalue of A and let B = (A I ) 1. Show that (a) the scalars j = 1/(j ), j = 1, . . . , n are the eigenvalues of B. (b) if xj is an eigenvector of B belonging to j , then xj is an eigenvector of A belonging to j . (c) if the power method is applied to B, then the sequence of vectors will converge to an eigenvector of A belonging to the eigenvalue that is closest to . [The convergence will be rapid if is much closer to one i than to any of the others. This method of computing eigenvectors by using powers of (A I ) 1 is called the inverse power method.]

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##### ISBN: 9780136009290

This full solution covers the following key subjects: . This expansive textbook survival guide covers 47 chapters, and 921 solutions. Since the solution to 6 from 7.6 chapter was answered, more than 213 students have viewed the full step-by-step answer. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. The answer to “Let A be an n n matrix with distinct real eigenvalues 1, 2, . . . , n. Let be a scalar that is not an eigenvalue of A and let B = (A I ) 1. Show that (a) the scalars j = 1/(j ), j = 1, . . . , n are the eigenvalues of B. (b) if xj is an eigenvector of B belonging to j , then xj is an eigenvector of A belonging to j . (c) if the power method is applied to B, then the sequence of vectors will converge to an eigenvector of A belonging to the eigenvalue that is closest to . [The convergence will be rapid if is much closer to one i than to any of the others. This method of computing eigenvectors by using powers of (A I ) 1 is called the inverse power method.]” is broken down into a number of easy to follow steps, and 149 words. The full step-by-step solution to problem: 6 from chapter: 7.6 was answered by , our top Math solution expert on 03/15/18, 05:24PM.

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