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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