1. Numerical Analysis – Eigenvalue and Eigenvector Hanyang University Jong-Il Park

2. Department of Computer Science and Engineering, Hanyang University Eigenvalue problem : eigenvalue x : eigenvector ※ spectrum: a set of all eigenvalue

3. Department of Computer Science and Engineering, Hanyang University Eigenvalue if det  x=0(trivial solution) To obtain a non-trivial solution, det  ;Characteristic equation

4. Department of Computer Science and Engineering, Hanyang University Properties of Eigenvalue 1) Trace 2) det 3) If A is symmetric, then the eigenvectors are orthogonal: 4) Let the eigenvalues of then, the eigenvalues of (A - aI)

5. Department of Computer Science and Engineering, Hanyang University Geometrical Interpretation of Eigenvectors Transformation : The transformation of an eigenvector is mapped onto the same line of. Symmetric matrix  orthogonal eigenvectors Relation to Singular Value if A is singular  0 {eigenvalues}

6. Department of Computer Science and Engineering, Hanyang University Eg. Calculating Eigenvectors(I) Exercise 1) ; symmetric, non-singular matrix ( = -1, -6) 2) ; non-symmetric, non-singular matrix ( = -3, -4)

7. Department of Computer Science and Engineering, Hanyang University Eg. Calculating Eigenvectors(II) 3) ; symmetric, singular matrix ( = 5, 0) 4) ; non-symmetric, singular matrix ( = 7, 0)

8. Department of Computer Science and Engineering, Hanyang University Discussion symmetric matrix => orthogonal eigenvectors singular matrix => 0 {eigenvalue} Investigation into SVD

9. Department of Computer Science and Engineering, Hanyang University Similar Matrices Eigenvalues and eigenvectors of similar matrices Eg. Rotation matrix

10. Department of Computer Science and Engineering, Hanyang University Similarity Transformation Coordinate transformation  x’=Rx, y’=Ry Similarity transformation  y=Ax  y’=Ry=RAx=RA(R -1 x’)= RAR -1 x’=Bx’ B = RAR -1

11. Department of Computer Science and Engineering, Hanyang University Numerical Methods(I) Power method  Iteration formula  for obtaining large

12. Department of Computer Science and Engineering, Hanyang University Eg. Power method

13. Department of Computer Science and Engineering, Hanyang University Numerical Methods (II) Inverse power method  Iteration formula  for obtaining small

14. Department of Computer Science and Engineering, Hanyang University Exploiting shifting property Let the eigenvalues of then, the eigenvalues of (A - aI) Finding the maximum eigenvalue with opposite sign after obtaining Accelerating the convergence when an approximate eigenvalue is available

15. Department of Computer Science and Engineering, Hanyang University Deflated matrices It is possible to obtain eigenvectors one after another Properly assigning the vector x is important Eg. Wielandt’s deflation

16. Department of Computer Science and Engineering, Hanyang University Eg. Using Deflation(I)

17. Department of Computer Science and Engineering, Hanyang University Eg. Using Deflation(II)

18. Department of Computer Science and Engineering, Hanyang University Numerical Methods (III) Hotelling's deflation method  Iteration formula: given  for symmetric matrices  deflation from large to small

19. Department of Computer Science and Engineering, Hanyang University Numerical Methods (IV) Jacobi transformation  Successive diagonalization without changing.  for symmetric matrices

20. Department of Computer Science and Engineering, Hanyang University Homework #7 Generate a 9x9 symmetric matrix A by using random number generator(Gaussian distribution with mean=0 and standard deviation=1.1]). Then, compute all eigenvalues and eigenvectors of A using the routines in the book, NR in C. Print the eigenvalues and their corresponding eigenvectors in the descending order.  You may use  jacobi(): Obtaining eigenvalues using the Jacobi transformation  eigsrt(): Sorting the results of jacobi() [Due: Nov. 19]