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Eigenvalues and Eigenvectors
Chapter 9 Burden and Faires 6/19/2019 Eigen Problems
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Symmetric Power Method
Using l2 norm speeds convergence when A is symmetric 6/19/2019 Eigen Problems
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Symmetric Power Method
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Convergence Criteria 6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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Power Method 6/19/2019 Eigen Problems
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Power Method 6/19/2019 Eigen Problems
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Power Method 6/19/2019 Eigen Problems
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Symmetric Power Method
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Symmetric Power Method
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Power Method Results 6/19/2019 Eigen Problems
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Symmetric Power Method Results 6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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Inverse Power Method Used to find eigenvalue of A that is closest to a specified number q. 6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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Apply Power Method 6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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6/19/2019 Eigen Problems
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m m(m) (x(m))T (1.0, 1.0, 1.0) T 1 (1.0, , ) T 2 (1.0, , ) T 3 (1.0, , ) T 4 (1.0, , ) T 5 (1.0, , ) T 6 (1.0, , ) T 6/19/2019 Eigen Problems
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More than One Eigenvector
Power Methods work for extracting one eigenvalue. If more than one eigenvalue is required then we resort. To deflation techniques to reduce n x n A matrix to an n-1 x n-1 matrix and repeat the process. 6/19/2019 Eigen Problems
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Theorem 9.15 6/19/2019 Eigen Problems
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Wielandt Deflation 6/19/2019 Eigen Problems
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Wielandt Deflation 6/19/2019 Eigen Problems
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Wielandt Deflation 6/19/2019 Eigen Problems
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Wielandt Deflation Matrix B’ is obtained by deleting ith row and ith column of A. Use power method to get l1 and v(1) Form B Use power method to get l2 and w (2) Use w(2) to get v(2) 6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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Example 6/19/2019 Eigen Problems
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Annihilation Technique
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Annihilation Technique
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Annihilation Technique
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Hotelling Deflation 6/19/2019 Eigen Problems
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Hotelling Deflation 6/19/2019 Eigen Problems
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