Lecture 13 - Eigen-analysis

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Lecture\'s Goals . Inverse Power MethodAccelerated Power MethodQR FactorizationHouseholder Hessenberg Method. Inverse Power Method. The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique.. Inverse Power Method. The algorithm is the same as the Power method and the eigenvector\" is not the eigenvector for the smallest eigenvalue. To obtain the smallest eigenvalue from the power method..
Lecture 13 - Eigen-analysis

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1. Lecture 13 - Eigen-analysis CVEN 302 June 27, 2001

2. Lecture?s Goals Inverse Power Method Accelerated Power Method QR Factorization Householder Hessenberg Method

3. Inverse Power Method

4. Inverse Power Method

5. Inverse Power Method

6. Example

7. Matlab Program

8. Accelerated Power Method

9. Accelerated Power Method

10. Example of Accelerated Power Method

11. Example of Power Method

12. Example of Accelerated Power Method

13. Example of Accelerated Power Method

14. Example of Accelerated Power Method

15. QR Factorization The technique can be used to find the eigenvalue using a successive iteration using Housholder transformation to find an equivalent matrix to [A] having an eigenvalues on the diagonal

16. QR factorization Another form of factorization A = Q*R Produces an orthogonal matrix (?Q?) and a right upper triangular matrix (?R?) Orthogonal matrix - inverse is transpose

17. QR factorization

18. QR Eigenvalue Method

19. QR Factorization Construction of QR Factorization Use Householder Reflections and Given Rotations to reduce certain elements of a vector to zero Use QR factorization that preserve the eigenvalues The eigenvalues of the transformed matrix are much easier to obtain

20. Jordon Canonical Form Any square matrix is orthogonally similar to a triangular matrix with the eigenvalues on the diagonal

21. Similarity Transformation Transformation of the matrix A of the form H-1AH is known as similarity transformation A real matrix Q is orthogonal if QTQ = I If Q is orthogonal, then A and Q -1AQ are said to be orthogonally similar The eigenvalues are preserved under the similarity transformation

22. Upper Triangular Matrix The diagonal elements Rii of the upper triangular matrix R are the eigenvalues

23. Householder Reflector Householder reflector is a matrix of the form It is straightforward to verify that Q is symmetric and orthogonal

24. Householder Matrix Householder matrix reduces zk+1 ,?,zn to zero To achieve the above operation, v must be a linear combination of x and ek

25. Householder Transformation

26. Householder matrix Corollary (kth Householder matrix): Let A be an nxn matrix and x any vector. If k is an integer with 1< k<n-1 we can construct a vector w(k) and matrix H(k) = I - 2w(k)w?(k) so that

27. Householder matrix Define the value ? so that The vector w is found by Choose ? = sign(xk)g to reduce round-off error

28. Householder Matrices

29. Example: Householder matrix

30. Example: Householder matrix

31. Basic QR Factorization [A] = [Q] [R] [Q] is orthogonal, QTQ = I [R] is upper triangular QR factorization using Householder matrices Q = H(1)H(2)?.H(n-1)

32. Example: QR Factorization

33. Similarity transformation B = QTAQ preserve the eigenvalues QR Factorization

34. Finding Eigenvalues Using QR Factorization Generate a sequence A(m) that are orthogonally similar to A Use Householder transformation H-1AH the iterates converge to an upper triangular matrix with the eigenvalues on the diagonal

35. QR Eigenvalue Method QR factorization: A = QR Similarity transformation: A(new) = RQ

36. Example: QR Eigenvalue

37. Example: QR Eigenvalue

39. Improved QR Method Using similarity transformation to form an upper Hessenberg Matrix (upper triangular matrix & one nonzero band below diagonal) More efficient to form Hessenberg matrix without explicitly forming the Householder matrices (not given in textbook)

41. Summary Single value eigen analysis Power Method Shifting technique Inverse Power Method QR Factorization Householder matrix Hessenberg matrix

42. Homework Check the Homework webpage

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