Some Engg. Applications of Matrices and Partial Derivatives

SUBMITTED TO:
DR. SONA RAJ MAM
SUBMITTED BY:
ANUGYAA SHRIVASTAVA (K12986)
BTECH CS 2ND SEM
SANJAY SINGH (K12336)
BTECH CE 2ND SEM
NISHANT YADAV (K12119)
BTECH CE 2ND SEM

PARTIAL DIFFERENTIATION
MATRIX AND DETERMINANTS
EIGEN VALUE AND EIGEN VECTORS

In engineering, it sometimes happens that the variation of one
quantity depends on changes taking place in two, or more,
other quantities. For example, the volume V of a cylinder is
given by V=πr2h. The volume will change if either radius r or
height h is changed. The formula for volume may be stated
mathematically as V=f (r, h) which means ‘V is some function
of r and h’.
Some other practical examples include:

(i) time of oscillation, t=2π√l/g i.e. t = f (l, g).
(ii) torque T =Iα, i.e. T =f (I, α).
(iii) pressure of an ideal gas p= mRT/V i.e. p=f (T, V).
When differentiating a function having two variables, one
variable is kept constant and the differential coefficient of
the other variable is found with respect to that variable. The
differential coefficient obtained is called a partial derivative
of the function.

Introduction to Matrices
You can view a matrix simply as a generalization of a vector, where we
arrange numbers in both rows and columns. Let's keep the number of
rows and columns arbitrary, letting m be the number of rows and n the
number of columns. We refer to such a matrix as an m×n matrix.
The arrangement of a matrix in rows and columns is more than just to
make it look pretty. The structure of a matrix allows us to define a
fundamental operation on matrices: multiplication. This multiplication
forms the basis of linear algebra. In particular, this matrix multiplication
allows matrices to represent linear transformations (or linear functions)
that transform vectors into other vectors. (A simple example of a linear
transformation is the rotation of a vector.) Other uses of matrices
involve calculating their determinant.

Introduction to Determinants
For any square matrix of order 2, we have found a
necessary and sufficient condition for invertibility. Indeed,
consider the matrix
The matrix A is invertible if and only if . We
called this number the determinant of A. It is clear from this,
that we would like to have a similar result for bigger
matrices (meaning higher orders). So is there a similar
notion of determinant for any square matrix, which
determines whether a square matrix is invertible or not?

In order to generalize such notion to higher orders, we will
need to study the determinant and see what kind of
properties it satisfies. First let us use the following notation
for the determinant

1. For any square matrix A, if A-λI=0 then the value of λ is
called eigen values of the matrix.
2. For each eigen value, if (A-λI)X=0, then the matrix X is
called the eigen vector of that eigen values λ.
3. Eigen values and eigen vectors are very important in
some engineering field .For example ,in civil engineering,
Eigen vector are used in mass calculation while making
structures in blocks.

Question.1 Find (a) AB (b)
BA by using matrix.
A 
3 2
1 0






B 
1 4
2 1






Solution.1
AB 
3 2
1 0






1 4
2 1






AB 
7 10
1 4






BA 
1 4
2 1






3 2
1 0






BA 
7 2
5 4







Question.2 Find the eigen values of 








51
122
A
)2)(1(23
12)5)(2(
51
122
2









 AI
two eigen values: 1,  2
Note: The roots of the characteristic equation can be
repeated. That is, λ1 = λ2 =…= λk. If that happens, the
eigen value is said to be of multiplicity k.
Solution.2

0
y
f
x
f
2
2
2
2






Laplace:
2
1jiji1ji
2
2
2
j1ijij1i
2
2
h
ff2f
y
f
h
ff2f
x
f  





 ,,,,,,
Substitute with the Central divided differences and assuming
that Dx = Dy = h
 ji1ji1jij1ij1i22
2
2
2
f4ffff
h
1
y
f
x
f
,,,,, 







  0f4ffff
h
1
y
f
x
f
ji1ji1jij1ij1i22
2
2
2






 ,,,,,
x
y Finite Difference Grid
i,ji-1,j i+1,j
i,j+1
i,j-1
-41 1
1
1
i,j
At i and j:

REAL WORLD
APPLICATION ON
MATRICES AND
DETERMINANTS

The Bermuda Triangle is a large triangular region in the
Atlantic ocean. Many ships and airplanes have been
lost in this region. The triangle is formed by imaginary
lines connecting Bermuda, Puerto Rico, and Miami,
Florida. Use a determinant to estimate the area of the
Bermuda Triangle.
E
W
N
S
Miami (0,0)
Bermuda (938,454)
Puerto Rico (900,-518)
.
.
.

The approximate coordinates of the Bermuda Triangle’s
three vertices are: (938,454), (900,-518), and (0,0). So
the area of the region is as follows:
Area  
1
2
938 454 1
900 518 1
0 0 1
Area  
1
2
[(458,884 0 0)(0 0 408,600)]
Area  447,242
Hence, area of the Bermuda Triangle is about 447,000
square miles.

There are two closed loops in the above circuit. loop 1:
e1, R1 and R3 and loop 2: e2, R2 and R3. e1 and e2 are
sources of voltages. R1, R2 and R3 are resistors. i1 is the
current flowing across R1 and i2 is the current flowing
across R2. We now apply Kirchhoff's law to each loop.
loop 1: e1 = R1 i1 + R3 (i1 - i2)
loop 2: e2 = R2 i2 + R3 (i2 - i1)

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 that 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

The above is a matrix equation that may be solved
using any known method to solve systems of
equations. Let e, R and i be matrices given by
The solution to the above matrix equation is given
by

where R -1 is the inverse matrix of R and is given by.

Graphic software uses matrix mathematics to process
linear transformations to render images. A square matrix,
one with exactly as many rows as columns, can represent
a linear transformation of a geometric object. For example,
in the Cartesian X-Y plane, the matrix reflects an object
in the vertical Y axis. In a video game, this would render
the upside-down mirror image of a castle reflected in a
lake.
If the video game has curved reflecting surfaces, such as a
shiny silver goblet, the linear transformation matrix would
be more complicated, to stretch or shrink the reflection.

REAL WORLD
APPLICATIONS ON
EIGEN VALUE AND
EIGEN VECTOR

1. Eigenvectors and eigen values are important for
understanding the properties of expander graphs,
which I understand to have several applications in
computer science (such as derandomizing random
algorithms). They also give rise to a graph partitioning
algorithm.
2. PRuler is an iPhone app that lets you measure objects
using a credit card and your iPhone camera. I'm told it
uses a Singular Value Decomposition (which is very
closely related to eigenvalues and eigenvectors)
Example 1

Example 2
Plot the sequence of
approximations of the maximum
eigenvector for the matrix
Starting with the initial vector:
These are shown in Figure 1. The
sequence of approximations is
shaded from blue to red. The last
plotted red vector is quite close to
the actual eigenvector of 9.54 ⋅
(0.763, 0.646)T (9.54 being the
corresponding eigen value).
Figure 1. The
sequence of
approximations of
the maximum
eigenvector with the
initial vector v = (-3.1,
5.2)T in black.

REAL WORLD APPLICATION ON PARTIAL
DIFFERENTIATION

Shape Processing using PDEs
Shape processing refers to operations such as denoising,
fairing, feature extraction, segmentation, simplification,
classification, and editing. Such operations are the basic
building blocks of many applications in computer graphics,
animation, computer vision, and shape retrieval.
Many shape processing operations can be achieved by
means of partial differential equationsor PDEs. The desired
operation is described as a (set of) PDE(s) that act on
surface information, such as area, normals, curvature, and
similar quantities. PDEs are a very attractive instrument:
They allow complex manipulations to be described precisely,
compactly, and measurably, and come with efficient and
effective numerical methods for solving them.
We present several applications of PDEs in shape
processing.

Partial Derivative in Economics:
In economics the demand of quantity and quantity supplied
are affected by several factors such as selling price,
consumer buying power and taxation which means there are
multi variable factors that affect the demand and supply. In
economics marginal analysis is used to find out or evaluate
the change in value of a function resulting from 1-unit
increase in one of its variables. For example Partial derivative
is used in marginal Demand to obtain condition for
determining whether two goods are substitute or
complementary. Two goods are said to be substitute goods if
an increase in the demand for either result in a decrease for
the other. While two goods are said to be complementary
goods if a decrease of either result in a decrease in the
demand. Example of complementary goods are mobile
phones and phone lines. If there is more demand
for mobile phone, it will lead to more demand for phone line
too.

In image processing edge detection algorithm is used which
uses partial derivatives to improve edge detection. Grayscale
digital images can be considered as 2D sampled points of a
graph of a function u(x,y) where the domain of the function is
the area of the image.
Partial Derivative in Engineering:

 From learning the applications of the eigen vectors and
eigen values, we came to know that the eigen vectors
and values is having very much importance in
engineering and in other fields also, but the thing is how
we connect that concept. Even these are very less
applications on eigen values and vectors and a lot of
work can be done on that because the innovation is
limitless.

REFERENCE BOOK
AUTHOR TITTLE PUBLICATION SITE
ERWIN
KREYSZIG
ADVANCED
ENGGINEERING
MATHEMATICS
JOHN WILEY &
SONS, INC.
www.wileyplus.com
JOHN BIRD HIGHER
ENGGINEERING
MATHEMATICS
JOHN BIRD.
PUBLISHED BY
ELSEVIER LTD.
www.elsevier.com
www.bookaid.org
www.sabre.org
JOHN BIRD ENGGINEERING
MATHEMATICS
LIBRARY OF
CONGRESS
CATALOGUING
IN PUBLICATION
www.books.elsevier.com
MICHAEL
BATTY
ESSENTIAL
ENGGINEERING
MATHEMATICS
DOVER
PUBLICATIONS
www.bookboon.com
H.K. DASS ADVANCED
ENGGINEERING
MATHEMATICS
S CHAND
PUBLICATIONS
https://www.schandpublish
ing.com

REFERENCE SITES
1. www.wileyplus.com
2. www.elsevier.com
3. www.sabre.org
4. http://math.stackexchange.com/questions/82413/practical-
applications-of-eigenvalues-eigenvectors-in-computer-
science
5. http://www.springer.com/in/book/9780857297839
6. https://www.chalmers.se/en/departments/cee/research/Grad
uate-education/graduate-courses-and-course-
descriptions/Pages/Mathematical-modelling-in-civil-
engineering-applications.aspx

Some Engg. Applications of Matrices and Partial Derivatives

Some Engg. Applications of Matrices and Partial Derivatives

More Related Content

What's hot (20)

Similar to Some Engg. Applications of Matrices and Partial Derivatives (20)

Recently uploaded (20)

Some Engg. Applications of Matrices and Partial Derivatives