# Matrix Tutorial

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```					       Matrix Tutorial

Transition Matrices
Graphs
Random Walks

Pádraig Cunningham
University College Dublin
2

Objective

   To show how some advanced mathematics has
practical application in data mining / information
retrieval.
   To show how some practical problems in data
mining / information retrieval can be solved using
matrix decomposition.
   To give you a flavour of some aspects of the course.
3

Stochastic Matrix: Markov process

   In 1998 (in some state) Land use is:
 30% I (Res), 20% II (Com), 50% III (Ind)
   Over 5 year period, the probabilities for change of use are:

From I     From II      From III

To I         0.8        0.1          0

To II        0.1        0.7          0.1

To III       0.1        0.2          0.9
4

Stochastic Matrix: Markov process

Land Use after 5 years

0.8           0.1       0                30
26        22        52          =
0.1           0.7       0.1              20

0.1           0.2       0.9              50

v1 = Av0
similarly
v2 = A2v0
and so on…
http://kinetigram.com/mck/LinearAlgebra/JPaisMatrixMult04/classes/JPaisMatrixMult04.html
5

Stochastic Matrix: Markov process

   When this converges:
    vn = Avn
 i.e.it converges to vn an eigenvector of A corresponding to
an eigenvalue 1.
 vn = [12.5 25 62.5]
6

Brief Review of Eigenvectors

   The eigenvectors v and eigenvalues λ of a matrix A
are the ones satisfying
   Avi = λivi

   i.e. vi is a vector that:
 Pre-multiplying        by matrix A
is the same as
 Multiplying        by the corresponding eigenvalue λi
7

The important property…
   Repeated application of the matrix to an arbitrary vector
results in a vector proportional to the eigenvector with largest
eigenvalue
   http://mathworld.wolfram.com/Eigenvector.html

n            n
lim A y = λ b v          1 1 1
n →∞
   What has this got to do with Random Walks?...

€
8

Transition Matrices & Random Walks

   Consider a random walk over a set of linked web
pages.
   The situation is defined by a transition (links) matrix.
   The eigenvector corresponding to the largest
eigenvalue of the transition matrix tells us the
probabilities of the walk ending on the various pages.
9

Web Pages Example

B
C
A

D
E
9

Web Pages Example
From
B                 A   B   C    D     E
A   1   0   0    0     1
C
B   1   1   0    0     0
A                To
C   0   1   1    0     1

D         D   0   0   1    1     0
E                 E   1   1   1    1     1
9

Web Pages Example
From
B                            A    B   C    D     E
A   1    0   0    0     1
C
B   1    1   0    0     0
A                              To
C   0    1   1    0     1

D                  D   0    0   1    1     0
E                            E   1    1   1    1     1

   Eigenvector corresponding to largest Eigenvalue
   0.38
   0.20
   0.49
   0.26
   0.71
9

Web Pages Example
From
B                                         A   B   C    D     E
A    1   0   0    0     1
C
B    1   1   0    0     0
A                                         To
C    0   1   1    0     1

D                       D    0   0   1    1     0
E                                        E    1   1   1    1     1

   Eigenvector corresponding to largest Eigenvalue
   0.38
   0.20
   0.49
   0.26
   0.71
   EVD: http://kinetigram.com/mck/LinearAlgebra/JPaisEVD04/classes/
JPaisEVD04.html
10

Review of Matrix Algebra

   Why matrix algebra now?
 The   Google PageRank algorithm uses Eigenvectors in
ranking relevant pages.
   Resources
 http://mathworld.wolfram.com/Eigenvector.html
 The Matrix Cookbook
11

Brief Review of Eigenvectors

   Eigenvectors are a special set of vectors associated with a
linear system of equations (i.e., a matrix equation).
   Each eigenvector is paired with a corresponding so-called
eigenvalue.
   The decomposition of a square matrix into eigenvalues and
eigenvectors is known as eigen decomposition

http://mathworld.wolfram.com/Eigenvector.html
12

Matrices in JAVA - e.g. JAMA

   Class EigenvalueDecomposition
 Constructor   EigenvalueDecomposition(Matrix Arg)
 Methods
   Matrix GetV()
   Matrix GetD()
   Where A is the original matrix and:
 AV=VD
13

Summary

   Data describing connections between objects can be
described as a graph
   This graph can be represented as a matrix
   Interesting structure can be discovered in this data
using Matrix Eigen-decomposition

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