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:

Matrix of Graphs Adjacency Matrix: If A(i, j) = 1: edge exists Else A(i, j) =

Laplacian of Graphs

Matrix of Weighted Graphs Weighted Matrix: If A(i, j) = w(i,j): edge exists Else A(i, j) = infty

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

Random walks Transition Matrix

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) –f ij 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) –f ij 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 interface computer user system response time EPS survey trees graph minors

Matrix in Image Processing

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

Random walks Transition Matrix