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Some ApplicAtionS of lineAr AlgebrA in computer Science
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Some ApplicAtionS of lineAr
AlgebrA in computer Science
R. C. Mittal
Department of Mathematics,
Jaypee Institute of Information Technology
Sector 62 NOIDA (U. P.)
R. C. Mittal
Applications of Linear
Algebra
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Page Ranking
We start with a question
How does the Google search web pages ???
Google searches web pages with the help of
“Page Ranking” for the given key word
searched for .
What is page ranking?
According to Wikipedia, page rank is an
algorithm used by Google Search to rank web
pages in their search engine results.
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• PageRank is a way of measuring the importance
of website pages.
• According to Google
PageRank works by counting the number and
quality of links to a page to determine a rough
estimate of how important the website is. The
underlying assumption is that more important
websites are likely to receive more links from
other websites.
Let us discuss a simple example
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Suppose there are 5 persons A, B, C , D and E.
Some of them know each other and consider
friends. Consider the following matrix
(1)
A=
This matrix indicate that A has all other 4 friends
and give equal weight to them,
B and D each has only one friend.
C has 3 friends and E has two friends.
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So friend score initially of X is row sum of A.
FS(A) = 11/6,
FS(B) = 13/12
FS(C) = ¼,
FS(D) = ¼
FS(E) = 19/12
Suppose we would like to know who will be
our real friend . Initially, you give equal score
to each one i.e. [1, 1, 1, 1, 1].
Now we try to iterate it as below
vn+1 = A vn
..(2)
with v0 = [1, 1, 1, 1, 1]T .
This is well known Power method.
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It will give dominant eigenvalue and corresponding
eigen vector.
Now let us replace the friend concept to link in a
web page . We form the matrix A as follows
• A webpage A is a friend of another webpage B if
it has a hyperlink to the webpage B.
• Google finds out who’s friends with who by just
crawling the web to find out who links to who.
• The number of “friends” a webpage has is just the
number of links present on the webpage.
• The matrix A is created by crawling the web. Once
we have it, we find its eigenvector to get an
estimate of the PageRanks.
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Perron Theorem: Let A = (aij ) be a real positive
square matrix. Then we have
(1) Let r=ρ(A), then r >0 and suppose it is the
largest eigen value.
(2)The eigenvalue r has a positive its
associated eigenvector.
(3) The eigenvalue r is a simple root of the
characteristic polynomial of A.
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Cipher Code Designing
Suppose we have to encode the text
“HEMA IS BEAUTIFUL ”
(2)
We select the linear transformation given by
the matrix
T=
(3)
The numerical equivalent of (2) is
8 5 13 1 27 9 19 27 2 5 1 21 20 9 6 21 12 27 (4)
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Now we apply transformation (3) on (4) in
groups of 3 values, to get
57 54 89 101 127 86 123 131 155 55 51 100 79 68 136 …
(5)
We convert this to modulo 27, to get
3 0 8 20 19 5 15 23 20 ….
(6)
So our cipher will be
C HTSEOWT …..
(7)
If we wish decipher it, we multiply (6) by the
inverse of T to get the original message
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Principal Component Analysis
Today a lot of data is generated daily in
different social sites. In fact 90% of today data
is generated in the last 3-4 years. This data is to
be properly analyzed to gather important and
relevant information.
PCA is a dimension reduction technique which
give most dominant features from the data.
These newly extracted features are known as
“Principal Components” .
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Key Points of PCA
• A principal component is a linear combination of
the original variables
• Principal components are extracted in such a way
that the first principal component explains
maximum variance in the dataset
• Second principal component tries to explain the
remaining variance in the dataset and is
uncorrelated to the first principal component
• Third principal component tries to explain the
variance which is not explained by the first two
principal components and so on
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It may be noted that each additional dimension
we add to the PCA technique captures less and
less of the variance in the model. The first
component is the most important one,
followed by the second, then the third, and so
on.
How to find Principal Components?
In order to obtain principal components, we
compute the singular values of data.
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Suppose A is a real mxn matrix, then singular
values of A are nothing but square roots of
non-negative eigen values of ATA . Here ATA is a
self adjoint matrix.
In singular value decomposition of the matrix A,
we decompose the matrix A into UDVT , U is a
mxm orthonormal matrix, V is a nxn
orthonormal matrix and D is a mxn matrix with
rxr diagonal matrix containing singular values
of A.
The matrix V gives us principal directions.
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Image Compression
We can store an image in a matrix where each
entry represents the value of its pixel. For
example, a gray scale image (black white
image) can be stored in PNG format where a
pixel has a value 0 or 1.
Suppose we have an image of size 480 x 423 i.e.
we have a matrix A of size 480 x 423 having
values only 0 or 1. We decompose it in SVD
form. Now we retain only first 30 singular
values and reconstruct the image.
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Qualitatively, we found very little difference in
the image.
However, in first case we have 480 x 423 =
203040 pixels values while after reduction we
have only 30 diagonal values , 480 x 30 values
in U and 30 x 423 values in VT .
Total values = 30(1 + 480 + 423) = 27120 value.
This is roughly 13% of original values.
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Face Recognition
• Suppose an organization has 1000 employees.
The organization has created a data base of
face images of the employees with 4 face
images at different angles of each employee. So
there are 4000 face images in SVD form of
employees.
Now whenever an employee comes to the
organization TV cameras are taking his photos
and try to match with stored images i.e. try to
match their singular values for face recognition
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References
1. G. Strang, Linear Algebra and its Applications, Thomson,
Brooks/Cole, 2006
2. R. K. Jain, S.R. K. Iyengar & M. K. Jain, Numerical
Methods, New Age International Publishers, New Delhi,
2004
3. M. H. Rafique, Under Standing Eigen-Vectors and Google
Page-Rank Algorithm,
https://medium.com/@muhammadhassaanrafique/unde
r-standing-eigen-vectors-and-google-page-rank-algorithm3ae77c2f92de
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