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Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang) Previous Lecture Eigenvalue and eigenvector The power method to find the dominant eigenvalue Today Inverse power method Deflation Find all eigenvalues of a symmetric matrix Inverse Power Method4.2 To find an eigenvalue other than the dominant one To derive the inverse power method, we will need the following result B is a polynomial of A With this theorem, we can transform an eigenvalue of A the dominant eigenvalue of B B=(A-qI)-1 The inverse power method © http://www.nurseweek.com/photos/98-7/questn1.gif Any question? How to find the eigenvalue smallest in magnitude? That’s Sections 4.2, the inverse power method So far, we can approximate The dominant eigenvalue of a matrix The one smallest in magnitude The one closest to some specified value Defaltion4.3 What if we need several of the largest/smallest eigenvalues? Deflation To remove an already determined solution, while leaving the remainder solutions unchanged Deflation x3-6x2+11x-6 =(x-1)(x2-5x+6) =(x-1)(x-2)(x-3) Deflation Within the context of polynomial rootfinding, we removed each root by dividing out the monomial Deflation For the matrix eigenvalue problem, we shift the previously determined eigenvalue to zero while leaving the remainder eigenvalues unchanged We need the relationship among the eigenvalues of a Matrix A and AT before the deflation algorithm Eigenvalues and eigenvectors of a matrix and its transpose The eigenvalues of AT are the same as those of A The eigenvectors from A and AT are orthogonal (associated with different eigenvalues) To shift the previously determined eigenvalue to zero Deflation For the matrix eigenvalue problem, we shift the previously determined eigenvalue to zero while leaving the remainder eigenvalues unchanged While leaving the remainder eigenvalues unchanged Deflation theorem © http://www.nurseweek.com/photos/98-7/questn1.gif Any question? The next question is, how to choose x for the formula B=A-1v1xT Wielandt Deflation Determine the two largest eigenvalues Like the power method, there is another deflation variation for symmetric matrices © http://www.nurseweek.com/photos/98-7/questn1.gif Any question? Section 4.4 Reduction from a symmetric matrix to its symmetric tridiagonal form Basic Idea To compute all of the eigenvalues of a symmetric matrix, we will proceed in two stages First Transform to symmetric tridiagonal form This step requires a fixed, finite number of operations (not iterative) Second An iterative procedure on the symmetric tridiagonal matrix that generates a sequence of matrices converged to a diagonal matrix Why Two Stages? (why not apply the iterative technique on the original matrix?) Section 4.4 focuses on the first stage Section 4.5 will consider the second stage Before going to the method, we need Two Concepts Similarity transformations Orthogonal matrices Similarity transformation A similarity transformation doesn’t affect the eigenvalues Orthogonal matrix Orthogonal Matrix A is symmetric and B=Q-1AQ=QTAQ BT=(QTAQ)T=QTAQ=B, that is, B is also symmetric Orthogonal Matrix A multiplication does not change the Euclidean norm (Qx)TQx=xTQTQx=xTx Since QTQ=I Now we can begin to reduce a symmetric matrix to tridiagonal form based on the use of Householder matrices Householder matrix Householder Matrices Are both symmetric and orthogonal (H-1=H) To reduce a symmetric matrix using Householder matrices which involves a sequence of n-2 similarity transformations Reduction idea Remember that we have not decided the w That’s Section 4.4, reduction to symmetric tridiagonal form