In this Presentation,Matrices with their Origin are explained along with its interdisciplinary Applications

1. Application Of Matrices
Prepared By: Shubham Mishr
M.Tech
IITRAM
Linear Algebra

3. Linear Transformations
So far we have seen a few linear transformations, but what makes them LINEAR?
To be linear, a transformation must have the following properties:
)v(T)u(T)vu(T

 For any vectors u and v in the domain of T
)u(cT)uc(T

 For all scalars c and every vector u in the domain of T
The basic idea is that for vector addition and scalar multiplication, the results are the
same if you perform the operation before or after you apply the transformation.
It is always the case that
0)0(T


This gives an easy way to test a transformation for linearity.
Also, a linear transformation always maps lines to lines (or to zero)

4. Linear Transformations
DEFINITIONS:
A mapping T:ℝn↦ℝm is said to be ONTO if each b in ℝm is the image
of at least one x in ℝn.
A mapping T:ℝn↦ℝm is said to be ONE-TO-ONE if each b in ℝm is the
image of at most one x in ℝn.
Domain
ℝn Range is
All of ℝm
T
Domain
ℝn
Range is a
subspace of ℝm
T
T is onto T is not onto
T
T is one-to-one
T
T is not one-to-one

5. Linear Transformations
A couple of quick tests to see if a transformation is one-to-one or onto:
More Columns than Rows – it can’t be One-to-One
More Rows than Columns – it can’t be Onto
More precisely:
A transformation is onto iff the columns of A span ℝm.
A transformation is one-to-one iff the columns are linearly independent.

6. For each x in ℝn , T(x) is computed as Ax, where A is an mxn matrix.
For simplicity, we denote such a matrix transformation by x↦Ax.
The domain of T is ℝn when A has n columns and the codomain of T is ℝm
when each column of A has m entries.
So an mxn matrix transforms vectors from ℝn into vectors from ℝm.
Here are a few examples of transformation matrices:
Matrix Transformations







10
21
A1











01321
53100
24121
A4











65
43
21
A3





 

11
11
A2

8.  Cryptography is the process of encrypting data so that third
party can’t read it and privacy can be maintained.
 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 by this Process, But there are
many other methods for cryptography
Application of Matrices

9.  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

10. 1 0 −1
 The encoding matrix be 0 1 0
0 −1 1
 Then, assign numeric value for “SUBMIT HER YOUR
PLANS” i.e. S=19, U=21, B=2, M=13, I=9, T=20,
space=27, H=8, E=5, R=18, space=27, Y=25, O=15, U=21,
R=18, space=27, P=16, L=12, A=1, N=14, S=19
Example: Let take the message
SUBMIT ME YOUR PLAN
S U B M I T H E R Y O U R P L A N S
19 21 2 13 9 20 27 8 5 18 27 25 15 21 18 27 16 12 1 14 19

11.  Since we are using a 3 by 3 matrix, we break the
enumerated message above into a sequence of 3 by 1
vectors:
[ ] [ ] [ ] [ ] [ ] [ ] [ ]19
21
2
13
9
20
27
27
8
5
21
1
14
19
18 15 27
16
25 18 12

12.  The message should be encoded by multiplying the above
3x1 matrix by the given encoding matrix.
19 13 27 18 15 27 1
21 9 8 27 21 16 14
2 20 5 25 18 12 19
1 0 −1
0 1 0
0 −1 1
This gives,
17 -7 22 -7 -3 15 -18
21 9 8 27 21 16 14
-19 11 -3 -2 -3 -4 5

13.  The columns of this matrix give the encoded message. The
message is transmitted in the following linear form
17, 21, -21, -7, 9, -9, 22, 8, -8, -7, 27, -27, -
3, 21, -21, 15, 16, -16, -18, 14, -14

14.  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 1=A, 2=B and so on and also 27=space.
Decryption Process

15.  The inverse of the encoding matrix should be taken
out such as:
1 1 1
0 1 0
0 1 1

16. Thus, to decode the message,
perform the matrix multiplication
and get the matrix
181 1 1 17 -7 22 -7 -3 15 -
0 1 0 21 9 8 27 21 16 14
0 1 1 -19 11 -3 -2 -3 -4 5
19 13 27 18 15 27 1
21 9 8 27 21 16 14
2 20 5 25 18 12 19

17.  The columns of this matrix, written in linear form,
give the original message:
S U B M I T H E R Y O U R P L A N S
19 21 2 13 9 20 27 8 5 18 27 25 15 21 18 27 16 12 1 14 19

18.  Many geologists make use of 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 Surveys

19.  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

21.  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.
Computer Animations

22. Since a digital image is basically a matrix to begin with: The rows
and columns of the matrix correspond to rows and columns of
pixels, and the numerical entries correspond to the pixels’ color
values. Decoding digital video, for instance, requires matrix
multiplication.
In the same way that matrix multiplication can help process digital
video, it can help process digital sound. A digital audio signal is
basically a sequence of numbers, representing the variation over
time of the air pressure of an acoustic audio signal. Many
techniques for filtering or compressing digital audio signals, such as
the Fourier transform, rely on matrix multiplication.

23. For Example:

24. Electronics
The behavior of many electronic components can be
described using matrices. Let A be a 2-dimensional
vector with the component's input voltage v1 and input
current i1 as its elements, and let B be a 2-dimensional
vector with the component's output voltage v2 and
output current i2 as its elements.
Then its behaviour can be described by B = H · A,
where H is a 2 x 2 matrix containing one impedance
element (h12), one admittance element (h21) and
two dimensionless elements (h11 and h22). Calculating a
circuit now reduces to multiplying matrices.

25. Also for Calculations:

26.  Matrices are very useful for organization, like for
scientists who have to record the data from their
experiments if it includes numbers.
 Stochastic matrices and Eigen vector solvers
are used in the page rank algorithms which are
used in the ranking of web pages in Google
search.
 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…

27. Other uses

28. One More…!!

29. References:
1)Slideshare.net
2)http://cseweb.ucsd.edu/~gill/CILASite/Resources/09Ch
ap5.pdf
3)http://mathworld.wolfram.com/LinearTransformation.
html
4)https://en.wikipedia.org/wiki/Linear_map
5)https://brilliant.org/wiki/linear-transformations/
6)Don’tmemorise.com