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

2. Determinants and Linear System Cramer’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. Matrices Representation of graphs
Adjacency matrix:

14. Adjacency Matrix: 1 5 2 3 4

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

16. Laplacian of Graphs 1 5 2 3 4

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

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

19. Random walks Transition Matrix1 2 3 4 5 6

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 1
interface 1 1
computer 1 1
user 1 1 1
system 1 1 2
response 1 1
time 1 1
EPS 1 1
survey 1 1
trees 1 1 1
graph 1 1 1
minors 1 1

26. Matrix in Image Processing

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

28. Random walks Transition Matrix1 2 3 4 5 6