Lecture 16 Cramer’s Rule, Eigenvalue and Eigenvector

1. Lecture 16Cramer’s Rule, Eigenvalue and Eigenvector Shang-Hua Teng

2. Determinants and Linear SystemCramer’s Rule

3. Cramer’s Rule • If det A is not zero, then Ax = b has the unique solution

4. Cramer’s Rule for Inverse Proof:

5. Where Does Matrices Come From?

6. Computer Science • Graphs: G = (V,E)

7. Internet Graph

9. View Internet Graph on Spheres

10. Graphs in Scientific Computing

11. Resource Allocation Graph

12. Road Map

13. Adjacency matrix: Matrices Representation of graphs

14. Adjacency Matrix: 1 5 2 3 4

15. 1 2 4 3 Matrix of Graphs Adjacency Matrix: • If A(i, j) = 1: edge exists Else A(i, j) = 0. 1 2 -3 4 3

16. Laplacian of Graphs 1 5 2 3 4

17. 1 2 4 3 Matrix of Weighted Graphs Weighted Matrix: • If A(i, j) = w(i,j): edge exists Else A(i, j) = infty. 1 2 -3 4 3

18. Random walks How long does it take to get completely lost?

19. 1 2 6 3 4 5 Random walks Transition Matrix

20. Markov Matrix • Every entry is non-negative • Every column adds to 1 • A Markov matrix defines a Markov chain

21. Other Matrices • Projections • Rotations • Permutations • Reflections

22. Term-Document Matrix • Index each document (by human or by computer) • fij counts, frequencies, weights, etc • Each document can be regarded as a point in m dimensions

23. Document-Term Matrix • Index each document (by human or by computer) • fij counts, frequencies, weights, etc • Each document can be regarded as a point in n dimensions

24. Term Occurrence Matrix

25. c1 c2 c3 c4 c5 m1 m2 m3 m4 human 1 0 0 1 0 0 0 0 0 interface 1 0 1 0 0 0 0 0 0 computer 1 1 0 0 0 0 0 0 0 user 0 1 1 0 1 0 0 0 0 system 0 1 1 2 0 0 0 0 0 response 0 1 0 0 1 0 0 0 0 time 0 1 0 0 1 0 0 0 0 EPS 0 0 1 1 0 0 0 0 0 survey 0 1 0 0 0 0 0 0 1 trees 0 0 0 0 0 1 1 1 0 graph 0 0 0 0 0 0 1 1 1 minors 0 0 0 0 0 0 0 1 1

26. Matrix in Image Processing

27. Random walks How long does it take to get completely lost?

28. 1 2 6 3 4 5 Random walks Transition Matrix