Question

The Inverse Power Method. Let $A$ be a nonsingular matrix. (a) Show that the eigenvalues of $A^{-1}$ are the reciprocals $1 / \lambda$ of the eigenvalues of $A$. How are the eigenvectors related? (b) Show how to use the Power Method on $A^{-1}$ to produce the smallest (in modulus) eigenvalue of $A$. (c) What is the rate of convergence of the algorithm? (d) Design a practical iterative algorithm based on the (permuted) $L U$ decomposition of $A$.

    The Inverse Power Method. Let $A$ be a nonsingular matrix. (a) Show that the eigenvalues of $A^{-1}$ are the reciprocals $1 / \lambda$ of the eigenvalues of $A$. How are the eigenvectors related? (b) Show how to use the Power Method on $A^{-1}$ to produce the smallest (in modulus) eigenvalue of $A$. (c) What is the rate of convergence of the algorithm?
(d) Design a practical iterative algorithm based on the (permuted) $L U$ decomposition of $A$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 9, Problem 6

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- Suppose $ \lambda $ is an eigenvalue of $ A $ with corresponding eigenvector $ v $, such that $ Av = \lambda v $. - To find the relationship with $ A^{-1} $, multiply both sides of the equation $ Av = \lambda v $ by $ A^{-1} $, yielding \( A^{-1}Av Show more…

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The Inverse Power Method. Let $A$ be a nonsingular matrix. (a) Show that the eigenvalues of $A^{-1}$ are the reciprocals $1 / \lambda$ of the eigenvalues of $A$. How are the eigenvectors related? (b) Show how to use the Power Method on $A^{-1}$ to produce the smallest (in modulus) eigenvalue of $A$. (c) What is the rate of convergence of the algorithm? (d) Design a practical iterative algorithm based on the (permuted) $L U$ decomposition of $A$.

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