Let A be an n × n matrix with distinct real eigenvaluesλ1, λ2, . . . , λn. Let λ be a scalar that is notan eigenvalue of A and let B = (A − λI)−1. Showthat 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 λ.
Let A be an n × n matrix with distinct real eigenvaluesλ1, λ2, . . . , λn. Let λ be a scalar that is notan eigenvalue of A and let B = (A − λI)−1. Showthat 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 λ.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 24EQ
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Let A be an n × n matrix with distinct real eigenvalues
λ1, λ2, . . . , λn. Let λ be a scalar that is not
an eigenvalue of A and let B = (A − λI)−1. Show
that if the power method is applied to B, then the
sequence of
of A belonging to the eigenvalue that is
closest to λ.
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