Applied Mathematics 3 Matrices.pdf

Applied Mathematics 3
Matrices Project Presentation By Prasad Baravkar pillai college of engineering

1. Dr. K. M. Vasudevan Pillai Campus, Plot No. 10, Sector 16, New Panvel East, Navi Mumbai,
Maharashtra 410206

AUTOMOBILE DEPARTMENT
DIRECT SECOND YEAR
PPT PRESENTATION OF MINI PROJECT

Title of the project : Matrices
Name of the guide : Suvarna Gaikwad
Name of the student : Prasad Sanjeev Baravkar Roll No. AEC303

2. MATRICES

3. INTRODUCTION
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols,
or expressions, arranged in rows and columns, which is used to represent a mathematical object
or a property of such an object.
For example,




Above example is a matrix with two rows and three columns. This is often referred to as a "two
by three matrix", a "2×3-matrix", or a matrix of dimension 2×3.

4. HISTORY
The term matrix was introduced by the 19th-century
English mathematician James Sylvester,
but it was his friend the mathematician
Arthur Cayley who developed the
algebraic aspect of matrices in
two papers in the 1850s.

Arthur Cayley
(1821-1895)

5. WHAT ARE MATRICES ?
•A matrix is defined as a rectangular array of numbers or symbols which are generally
arranged in rows and columns.

•The order of the matrix can be defined as the number of rows and columns.

•The entries are the numbers in the matrix known as an element.

•The plural of matrix is matrices.

•The size of a matrix is denoted as ‘n by m’ matrix and is written as m×n, where n= number
of rows and m= number of columns.

6. TYPES OF MATRIX

1. Row matrix

2.Column matrix

3.Square matrix

4.Diagonal matrix



5.Upper triangular matrix

6.Lower triangular matrix

7.Unit matrix

8.Zero matrix

7. BASIC DEFINATIONS
•Row Matrix : A matrix having only one row is called as row matrix (or row vector)
General form of row matrix is A = [ a
11
, a
12
, a
13
,……, a
n
]

•Column Matrix : A matrix having only one column is called as Column matrix. (or column vector)

Column matrix is in the form of A =


•Square Matrix : A matrix in which number of rows & columns are equal is called as a square
matrix

General form of square matrix is A =

8. BASIC DEFINATIONS
•Diagonal Matrix : A square matrix [a
ij
]
n
is said to be a diagonal matrix if a
ij
= 0 for i ≠
j. (i.e., all the elements of the square matrix other than diagonal elements are zero).

•Upper Triangular Matrix : A = [a
ij
]
m × n
is said to be upper triangular, if a
ij
= 0 for i >
j.(i.e. all the elements below the diagonal elements are zero).

•Lower Triangular Matrix : A = [a
ij
]
m × n
is said to be a lower triangular matrix, if a
ij
=
0 for i < j. (i.e. all the elements above the diagonal elements are zero.)

9. BASIC DEFINATIONS
•Unit Matrix : Unit matrix is a diagonal matrix in which all the diagonal elements are unity.
Unit matrix of order ‘n’ is denoted by I
n
(or I)
i.e. A = [a
ij
]
n
is a unit matrix when a
ij
= 0 for i ≠ j & a
ii
= 1


•Zero Matrix : A = [a
ij
]
m × n
is called a zero matrix, if a
ij
= 0 i & j.

10. •SCALAR MATRIX
Scalar matrix is a diagonal matrix in which all the diagonal elements are same
A = [a
ij
]
n
is a scalar matrix, if (i) a
ij
= 0 for i ≠ j and
(ii) a
ij
= k for i = j.
•MULTIPLICATION OF MATRIX BY SCALAR
Let λ be a scalar (real or complex number) & A = [a
ij
]
m × n
be a matrix. Thus the product λ A
is defined as A = [b
ij
]
m × n
where b
ij
= λ a
ij
∀ i & j.
Note : If A is a scalar matrix, then A = λ I, where is the diagonal element.

11. •Addition of Matrices
Let A and B be two matrices of same order (i.e. comparable matrices). Then A + B is defined
to be. A + B = [a
ij
]
m × n
+ [b
ij
]
m × n.

= [c
ij
]
m × n
where c
ij
= a
ij
+ b
ij
∀ i & j.

•Subtraction of Matrices
Let A & B be two matrices of same order.
Then A – B is defined as A + – B where – B is (– 1) B.

12. Eigenvalue and Eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero
vector that changes at most by a scalar factor when that linear transformation is applied to it. The
corresponding eigenvalue, often denoted by λ (lambda) , is the factor by which the eigenvector is
scaled.

Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in
which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched.
If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional
vector space, the eigenvector is not rotated.

13. Eigenvalue Problem
The Eigenvalue Problem Consider a n x n matrix A

❑Vector equation: Ax = λ x
•Seek solutions for x and λ
•λ satisfying the equation are the eigenvalues
•Eigenvalues can be real and/or imaginary; distinct and/or repeated
•x satisfying the equation are the eigenvectors

❑Nomenclature
•The set of all eigenvalues is called the spectrum
•The largest of the absolute values of the eigenvalues is called the spectral radius.

14. Determining Eigenvalues
❑Vector equation
•Ax = λx (A-λΙ)x = 0
•A-λΙ is called the characteristic matrix
•Non-trivial solutions exist if and only if:



•This is called the characteristic equation

❑Characteristic polynomial
•nth-order polynomial in λ
•Roots are the eigenvalues {λ
1
, λ
2
, …, λ
n
}

15. Eigenvalue Properties
•Eigenvalue Properties Eigenvalues of A and AT are equal
•Singular matrix has at least one zero eigenvalue
•Eigenvalues of A-1 : 1/λ1, 1/λ2, …, 1/λn
•Eigenvalues of diagonal and triangular matrices are equal to the diagonal elements

•Trace


•Determinant

16. Eigenvalue Example

•Characteristic Matrix



•Characteristic Equation



•Eigenvalues

17. Determining Eigenvectors
•First determine eigenvalues: {λ1, λ2, …, λn}
•Then determine eigenvector corresponding to each eigenvalue:
•Eigenvectors determined up to scalar multiple Distinct eigenvalues
» Produce linearly independent eigenvectors
•Repeated eigenvalues
» Produce linearly dependent eigenvectors
» Procedure to determine eigenvectors more complex (see text)

18. Eigenvector Example

•Eigenvalues


•Determine Eigenvectors : Ax = λx


•Eigenvector for λ
1
= -5


•Eigenvector for λ
1
= 2

19. Application Matrices
⬗Computer Graphics

⬗Optics

⬗Cryptography

⬗Economics

⬗Chemistry

⬗Geology

⬗Robotics and animation

⬗Wireless communication and signal
processing

⬗Finance ices

⬗Mathematics

20. ⬗Use of Matrices In Computer Graphics
•Architecture, cartoons, automation were done by using computer graphics.
•Used to project three dimensional images into two dimensional planes in the field of graphics.
•In Graphics, digital image is treated as a matrix to start 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.
•Video game graphics Matrices are also used to express graphs.

21. ⬗Use of Matrices in Cryptography
•Using Cryptography only the relevant person can get data and relate information.
•Encrypting the video signal.
•A digital audio or video signal is firstly taken as a sequence of numbers representing the
variation over time of air pressure of an acoustic audio signal. The filtering techniques are used
which depends on matrix multiplication.

22. ⬗Use of Matrices in Wireless Communication
•Model the wireless signals and to optimize them.
•For detection, extractions and processing of the information.
•Signal estimation and detection problems.
•They are used in sensor array signal processing and design of adaptive filters.
•Representing digital.

23. ⬗Use of Matrices in Science
•Use for reflection and for refraction.
•Useful in electrical circuits and quantum mechanics and resistor conversion of electrical
energy.
•Matrices are used to solve AC network equations in electric circuits.

24. ⬗Application of Matrices in Mathematics
•Solving linear equations.
•Matrices are incredibly useful things that happen in many various applied areas.
• Application of matrices in mathematics applies to many branches of science, also as different
mathematical disciplines.
•Engineering Mathematics is applied in our daily life.

25. Examples with Solutions
Example 1 Example 2

26. ⬗Eigenvalues and Eigenvector Example and Solutions

27. Conclusion
⬗From this project we learned about the different types of matrices and why we
are actually using it.
⬗The matrix are useful and powerful in the mathematical analysis and collecting
data. Besides the simultaneous equations, the characteristic of the matrices are
useful in the programming where we putting in array that is a matrix also to
store the data.