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1. TN 112
Lecture 5
Applications of Matrix

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There are many uses of matrix from science and engineering
fields to economic and business fields. As an example, matrix
can be used in
Computer Animations and Graphics
Wireless Communication
To write secret messages using Matrix
Seismic Survey
Google Search
Economics
Cryptography
Physics
Broad Applications of Matrix

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Matrix transforms are very useful within the world of computer graphics.
Software and hardware graphics processor uses matrices for performing
operations such as scaling, translation, reflection and rotation.
A square matrix is a simple way to represent linear object transformations.
In the realm of graphics, matrices are used to project three-dimensional
images into two-dimensional planes. To begin with, a digital image is
treated as a matrix in graphics. The matrix’s rows and columns
correspond to the rows and columns of pixels, and the numerical entries
correspond to the colour values of the pixels. In video game graphics,
manipulating a point with matrices is a typical mathematical strategy.
Graphs are also expressed using matrices.
Computer Animations and Graphics

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Wireless signals are modelled and optimized using matrices.
Matrixes are used to detect, extract, and process the
information encoded in signals. The estimation of signals and
detecting problems on wireless communication heavily relies
on matrices.
We all know that a significant part of the telecommunication
business is wireless communication. We can find the
application of matrices over here, as it is useful for processing
and displaying digital images.
Matrices are used to cover channels, hidden tent within web
pages, hidden files in plain sight, null ciphers and
steganography. In recent wireless internet connection through
mobile phone, known as wireless application protocol (wap)
also utilize matrices in the form of stenography.
In Wireless Communication

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 Word games and mathematical puzzles often center on
codes and secret messages. But secret messages aren't
just for fun and games ; they're used all over the world, and
in all kinds of circumstances.
 Can you think of a code which was designed to send
messages by telegraph, using sequences of short and long
tones called dots and dashes? The Matrix Code is a
method for creating and decoding secret messages.
To write secret messages using Matrix

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 Many geologists make use certain types of matrices for
seismic surveys.
 The seismic survey is one form of geophysical survey that
aims at measuring the earth’s (geo-) properties by means
of physical (-physics) principles such as magnetic, electric,
gravitational, thermal, and elastic theories.
Seismic Survey

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 Matrices in google search Stochastic matrices and Eigen
vector solver in the page rank algorithms which are used in
the ranking of page of Google search
In Google Seach

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 Matrices are used to calculate gross domestic product in
economics, and help in calculation for producing goods
more efficiently.
 It is seen that through input- output analysis that is used in
matrix, a researcher can get information about what level of
output should be of each industry at the existing
technology.
In Economics

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What does cryptography mean?
Cryptography, is concerned with keeping communications
private.
Cryptography mainly consists of Encryption and Decryption
Encryption is the transformation of data into some unreadable
form.
 Its purpose is to ensure privacy by keeping the
information hidden from anyone for whom it is not
intended, even those who can see the encrypted
data.
Decryption is the reverse of encryption It is the
transformation of encrypted data back into some intelligible
form.
Cryptography

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 Encryption and Decryption require the use of some secret
information, usually referred to as a key.
 Depending on the encryption mechanism used, the same
key might be used for both encryption and decryption, while
for other mechanisms, the keys used for encryption and
decryption might be different.
 Cryptography includes techniques such as merging words
with images, and other ways to hide information in storage
or transit.
 Cryptography involves encrypting data so that a third party
cannot intercept and read the data. In this, cryptography
matrix is a must which is very essential in engineering.
Cryptography

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Cryptography
It was started with the TV cable industries where even
people who were not the customer could watch the TV
programs
 So, Videocipher encryption system was invented which
would convert signals into digital form i.e. encrypt it, and the
data were send over the satellite. The Videocipher box would
decrypt the signal and those satellite dish owner who had
Videocipher box would receive the decrypted signal i.e. the
original signal before encryption.
 In matrix same thing can be done.
 How?
 See the next slide…
Typical examples: Cryptography

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 First, write a numerical value
for each letter i.e. A=1, B=2,
and Z=26, and space=27.
 The data should be placed in
matrix form i.e. in 2x1 or 3x1
matrix form.
 The data should be multiplied
by given encoding matrix.
 Then, write the answer
(value after multiplying) in
linear form.
How to encrypt data? Encryption process

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Example 1: Encryption process

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We then encode the matrix by multiplying the 3 by 1 matrix
with the given encoding matrix as follows:
Example 1: Encryption process

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The columns of the resulting matrix gives the encoded
message. The message is then transmitted in the linear form
as:
17, 21, -19, -7, 9, 11, 22, 8, -3, -7, 27, -2, -3, 21, -3, 15, 16, -4,
-18, 14, 5
Example 1: Encryption process

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 The encrypted number should be written in matrix form.
 The inverse of the encoding matrix should be found.
 Multiply the inverse encoding matrix, i.e. decoding matrix
with the encrypted number.
 Write the answer in linear form.
 Assign the numbers with the corresponding letters i.e. 1=A,
2=B and so on and also 27=space.
Example 1: Decryption process

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Example 1: Decryption process

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 Write the answer in linear form as:
19 21 2 13 9 20 27 8 5 18 27 25 15 21 18 27 16 12 1 14 19
 Assign the numbers with the corresponding letters i.e. 1=A,
2=B and so on and also 27=space.
i.e.
19 21 2 13 9 20 27 8 5 18 27 25 15 21 18 27 16 12 1 14 19
S U B M I T H E R Y O U R P L A N S
Hence, the message received is SUBMIT HER YOUR PLANS
Example 1: Decryption process

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 Production Scheduling
Labor and material costs for manufacturing two guitar models
are given in the table below: Suppose that in a given week
$1800 is used for labor and $1200 used for materials. How
many of each model should be produced to use exactly
each of these allocations?
Example 2: Economics

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Example 2: Solution

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Example 2: Solution

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In physics related applications, matrices are applied in the
study of electrical circuits, quantum mechanics and optics.
eg. Solving a system of linear equations in electrical circuit.
Consider the circuit below:
Matrix Applications In Physics
Kirchhoff's law to each loop.
loop 1: e1 = R1 i1 + R3 (i1 - i2)
loop 2: e2 = R2 i2 + R3 (i2 - i1)

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Question: If e1, e2, R1, R2 and R3 are known, how do you
calculate i1 and i2?
This circuit is simple and involves only two equations.
However electric circuits can be much more complicated than
the one above and matrices are suitable to answer the above
question. Let us group like terms in the above system of
equations
e1 = i1 (R1 + R3) - i2 R3
e2 = - i1 R3 + i2(R2 + R3)
and then write it in matrix form as follows
Matrix Applications In Physics

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Matrix Applications In Physics

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Example 3: Electrical Circuit

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Example 3: Electrical Circuit

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Try: Cryptography

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 Matrices are very useful for organization, like for scientists
who have to record the data from their experiments if it
includes numbers.
 In engineering, math reports are recorded using matrices.
 And in architecture, matrices are used with computing. If
needed, it will be very easy to add the data together, like
with matrices in mathematics.
Other uses….

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END!!