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

Determinants and Linear System Cramer’s Rule

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

Cramer’s Rule for Inverse Proof:

Where Does Matrices Come From?

Computer Science Graphs: G = (V,E)

Internet Graph

View Internet Graph on Spheres

Graphs in Scientific Computing

Resource Allocation Graph

Road Map

Matrices Representation of graphs Adjacency matrix:

Adjacency Matrix: 1 5 2 3 4

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

Laplacian of Graphs 1 5 2 3 4

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

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

Random walks Transition Matrix 1 2 3 4 5 6

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

Other Matrices Projections Rotations Permutations Reflections

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

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

Term Occurrence Matrix

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

Matrix in Image Processing

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

Random walks Transition Matrix 1 2 3 4 5 6