Matrices and Its Application

5/28/2018 Matrices and Its Application

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Term Paper

On

Matrices and its Application


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Table of content

1.1 Introduction.

1.2 Background of the Study.

1.3 Objective of the report.

1.4 Methodology..

1.5 Scope and limitation of the study..

2.1 Definition of Matrix

2.2 Matrix Notation.

2.3 History of matrix.

2.4 Types of matrix..

2.4.1 Row Matrix

2.4.2 Column Matrix

2.4.3 Rectangular Matrix

2.4.4 Square Matrix..

2.4.5 Zero Matrix..

2.4.6 Upper Triangular Matrix.

2.4.7 Lower Triangular Matrix

2.4.8 Diagonal Matrix

2.4.9 Scalar Matrix.

2.4.10 Identity Matrix

2.4.11 Transpose Matrix

2.4.12 Regular Matrix

2.4.13 Singular Matrix

2.4.14 Idempotent Matrix

2.4.15 Involutive Matrix

2.4.16 Symmetric Matrix

2.4.17 Antisymmetric Matrix

2.4.18 Orthogonal Matrix

3.1. Properties of matrix operation

3.1.2 Properties of subtraction

3. 1.1 Properties of Addition

3.1.3 Properties of Matrix Multiplication

3.1.4 Properties of Scalar Multiplication..

3.1.5 Properties of the Transpose of a Matrix..


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3.2 Matrix Operation

3.2.1 Matrix

Equality

3.2.2 Matrix Addition

3.2.3 Matrix Subtraction

3.2.4 Matrix Multiplication

3.2.5 Scalar Matrix Multiplication

3.2.6 Matrix Inverse

3.3 Elementary Matrix Operations

3.3.1 Elementary Operations

3.3.2 Elementary Operation Notation

4.1 Application of Matrix4.1.1 Solving Linear Equations

4.1.2 Electronics

4.1.3 Symmetries and transformations in physics

4.1.4 Analysis and geometry

4.1.5 Probability theory and statistics

4.1.6 Cryptography

4.2. Application of Matrices in Real Life

5.1 Findings

5.2 Recommendation

5.3 Conclusion


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Executive Summary

Matrices are one of the most powerful tools in Mathematics. We have prepared thisreport, Matrices and its Application, to describe about matrices and its application in

our life.

The origins of mathematical matrices lie with the study of systems of simultaneous linear

equations. An important Chinese text from between 300 BC and AD 200,Nine Chapters

of the Mathematical Art (Chiu Chang Suan Shu ), gives the first known example of the

use of matrix methods to solve simultaneous equations.

Matrices have been using widely in various sectors of modern life. Matrices are used in

inventory model, electrical networks, and other real life situations. In mathematics, a

matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows andcolumns. The individual items in a matrix are called its elements or entries. Matrix is

used quite a bit in advanced statistics.

In our study we have focused various types of matrices in chapter two. These are Row

matrix, Column matrix, Rectangular matrix, Square matrix, Diagonal matrix, Identity

matrix, Transpose matrix etc.

In chapter three we have discussed about the matrices properties of addition, subtraction,

multiplication, transpose. The study also explores the matrices operations, elementary

Matrix operations.The study also covers the application of Matrices in Mathematics and real life in

different areas of business and science like budgeting, sales projection, cost estimation

etc.

The report ends with some findings of analysis and recommendations regarding its

applications.
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CHAPTER-1

INTRODUCTION


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1.1 Introduction

In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or

expressions, arranged in rows and columns. The individual items in a matrix are called

its elements or entries. An example of a matrix with 2 rows and 3 columns is

Matrices of the same size can be added or subtracted element by element. But the rule for

matrix multiplication is that two matrices can be multiplied only when the number of

columns in the first equals the number of rows in the second. A major application of

matrices is to represent linear transformations. Another application of matrices is in the

solution of a system of linear equations.

Applications of matrices are found in most scientific fields. In every branch of physics,

including classical mechanics, optics, electromagnetism, quantum mechanics, and

quantum electrodynamics, they are used to study physical phenomena, such as the

motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional

image onto a 2-dimensional screen.

In probability theory and statistics, stochastic matrices are used to describe sets of

probabilities; for instance, they are used within the PageRank algorithm that ranks the

pages in a Google search. Matrix calculus generalizes classical analytical notions such as

derivatives and exponentials to higher dimensions.

A major branch of numerical analysis is devoted to the development of efficient

algorithms for matrix computations, a subject that is centuries old and is today anexpanding area of research.

Matrix decomposition methods simplify computations, both theoretically and practically.

Algorithms that are tailored to particular matrix structures, such as sparse matrices and

near-diagonal matrices, expedite computations in finite element method and other

computations. Infinite matrices occur in planetary theory and in atomic theory. A simple

example of an infinite matrix is the matrix representing the derivative operator, which

acts on the Taylor series of a function.


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1.2 Background of the Study

It is an opportunity for the students to acquire an in-depth knowledge through the term

paper preparing. The world is rapidly changing with new innovations in almost every

discipline. We should use the faster calculation and solution tools to solve the problem of

different fields. Matrix is a tool which we can apply in both mathematics and other

sciences. This mathematical tool simplifies our work to a great extent when compared

with other straight forward method. Some of the merely take advantage of the compact

representation of a set of numbers in a matrix. Matrices are a key tool in linear algebra,

one uses of matrices is to represent linear transformation Matrices can also keep track of

the coefficients in a system of linear equations.

We choose to do the study for the reason that in doing that in doing so. We can solve

linear transformations and transition by matrix operation.

1.3 Objective Of The Report

The main objective of education is to acquire knowledge. There are two types of

objectives of the report. One is primary objective and the other is Secondary objective.

Primary Objective:

The primary objective of this report is to use the theoretical concepts, gained in the

classroom situations, in analyzing real life scenarios. So that it adds value to the

knowledge base of us. This is also a partial requirement of the fulfillment of the course.

Secondary Objectives:

The secondary objectives are as follows:

To know the basic concept of matrices

To know the different operation of matrices

To know the historical background of matrices

To know the properties of Matrix operations.

To know the different application of matrices


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1.4 Methodology

The study is based on secondary data. The source of secondary data have been processed

and analyzed systematically.

Sources of Secondary data:

Text Books

Class Materials

Different report and research paper

Different websites

1.5 Scope and Limitation of the Study

The study focuses on the basics of matrices and the use of matrices. This paper also

emphasizes on the uses of matrix in different field like in science, engineering,

accounting, economics, inventory, business etc.

During the completion of this term paper following limitations of the study can be

mentioned.

Time frame for the study was very limited.

Lack of available information for making comprehensive study

Lack of experiences has acted as constraints in the way of study

Lack of group study to complete the paper

No primary data are considered

It seems to us that this report is as a study report based on the existing information

available on the topic Matrices and its Application


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CHAPTER- 2

THEORITICAL OVERVIEW


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2.1 Definition of MatrixA matrixis a rectangular array ofnumbers or other mathematical objects, for which

operations such asaddition and multiplication are defined. Most commonly, a matrix over

afield Fis a rectangular array of scalars from F. Most of this article focuses on realand complex

matrices, i.e., matrices whose elements arereal numbers orcomplex numbers, respectively.

More general types of entries are discussedbelow.For instance, this is a real matrix:

The numbers, symbols or expressions in the matrix are called its entriesor its elements. The

horizontal and vertical lines of entries in a matrix are called rowsand columns, respectively.

2.2Matrix Notation

Matrices are commonly written in box brackets:

An alternative notation uses largeparentheses instead ofbox brackets:

The specifics of symbolic matrix notation varies widely, with some prevailing trends. Matrices

are usually symbolized using upper-case letters (such as A in the examples above), while the

correspondinglower-case letters, with two subscript indices (e.g., a11, or a1,1), represent the

entries. In addition to using upper-case letters to symbolize matrices, many authors use a

specialtypographical style, commonly boldface upright (non-italic), to further distinguish

matrices from other mathematical objects. An alternative notation involves the use of a double-

underline with the variable name, with or without boldface style, (e.g., ).

The entry in the i-th row andj-th column of a matrix A is sometimes referred to as the i,j, (i,j), or

(i,j)th

entry of the matrix, and most commonly denoted as ai,j, or aij. Alternative notations for

that entry areA[i,j] orAi,j. For example, the (1,3) entry of the following matrix A is 5 (also

denoted a13, a1,3,A[1,3] orA1,3):
http://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Basic_operationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_multiplicationhttp://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Matrix_(mathematics)#More_general_entrieshttp://en.wikipedia.org/wiki/Parenshttp://en.wikipedia.org/wiki/Box_brackethttp://en.wikipedia.org/wiki/Upper-casehttp://en.wikipedia.org/wiki/Lower-casehttp://en.wikipedia.org/wiki/Emphasis_(typography)http://en.wikipedia.org/wiki/Emphasis_(typography)http://en.wikipedia.org/wiki/Lower-casehttp://en.wikipedia.org/wiki/Upper-casehttp://en.wikipedia.org/wiki/Box_brackethttp://en.wikipedia.org/wiki/Parenshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#More_general_entrieshttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Basic_operationshttp://en.wikipedia.org/wiki/Number


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Sometimes, the entries of a matrix can be defined by a formula such as ai,j=f(i,j). For example,

each of the entries of the following matrix A is determined by aij= ij.

In this case, the matrix itself is sometimes defined by that formula, within square brackets or

double parenthesis. For example, the matrix above is defined as A = [ i-j], or A = ((i-j)). If matrix

size is m n, the above-mentioned formulaf(i,j) is valid for any i= 1, ..., mand anyj= 1, ..., n.

This can be either specified separately, or using m nas a subscript. For instance, the

matrix Above is 3 4 and can be defined as A = [ij] (i= 1, 2, 3;j= 1, ..., 4), or A = [ij]34.

Some programming languages utilize doubly subscripted arrays (or arrays of arrays) torepresent an m--nmatrix. Some programming languages start the numbering of array indexes

at zero, in which case the entries of an m-by-nmatrix are indexed by 0 im 1and 0 jn

1. This article follows the more common convention in mathematical writing where

enumeration starts from 1.

Theset of all m-by-nmatrices is denoted (m, n).

2.3 History of matrix

The origins of mathematical matrices lie with the study of systems of simultaneous linear

equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the

Mathematical Art (Chiu Chang SuanShu ), gives the first known example of the use of matrix

methods to solve simultaneous equations.

In the treatise's seventh chapter, "Too much and not enough," the concept of a determinant

first appears, nearly two millennia before its supposed invention by the Japanese

mathematicianSeki Kowa in 1683 or his German contemporaryGottfried Leibnitz (who is also

credited with the invention of differential calculus, separately from but simultaneously with

Isaac Newton).

More uses of matrix-like arrangements of numbers appear in chapter eight, "Methods of

rectangular arrays," in which a method is given for solving simultaneous equations using a

counting board that is mathematically identical to the modern matrix method of solution

outlined byCarl Friedrich Gauss (1777-1855), also known asGaussian elimination .

The term "matrix" for such arrangements was introduced in 1850 byJames Joseph Sylvester .

Sylvester, incidentally, had a (very) brief career at the University of Virginia, which came to an

abrupt end after an enraged Sylvester hit a newspaper-reading student with a sword stick and

fled the country, believing he had killed the student!
http://en.wikipedia.org/wiki/Set_(mathematics)http://saxakali.com/COLOR_ASP/developcm3.htmhttp://saxakali.com/COLOR_ASP/developcm3.htmhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Seki.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Leibniz.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Gauss.htmlhttp://www.sosmath.com/matrix/system1/system1.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Sylvester.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Sylvester.htmlhttp://www.sosmath.com/matrix/system1/system1.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Gauss.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Leibniz.htmlhttp://www-history.mcs.st-and.ac.uk/history/Mathematicians/Seki.htmlhttp://saxakali.com/COLOR_ASP/developcm3.htmhttp://en.wikipedia.org/wiki/Set_(mathematics)


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Since their first appearance in ancient China, matrices have remained important mathematical

tools. Today, they are used not simply for solving systems of simultaneous linear equations, but

also for describing thequantum mechanics of atomic structure, designing computergame

graphics , analyzingrelationships , and even plotting complicateddance steps !

The elevation of the matrix from mere tool to important mathematical theory owes a lot to thework of female mathematicianOlga Taussky Todd (1906-1995), who began by using matrices to

analyze vibrations on airplanes during World War II and became the torchbearer for matrix

theory.

2.4 Types of Matrices

2.4.1 Row Matrix

A row matrix is formed by a single row.

2.4.2 Column Matrix

A column matrix is formed by a single column.

2.4.3 Rectangular Matrix

A rectangular matrix is formed by a different number of rows and

columns, and its dimension is noted as: mxn .

2.4.4 Square Matrix

A square matrix is formed by the same number of rows and columns.

The elements of the form a i iconstitute the principal diagonal.

The secondary diagonal is formed by the elements with i+j = n+1.
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2.4.5 Zero Matrix

In a zero matrix, all the elements are zeros.

2.4.6Upper Triangular Matrix

In an upper triangular matrix, the elements located below the

diagonal are zeros.

2.4.7Lower Triangular Matrix

In a lower triangular matrix, the elements above the diagonal are

zeros.

2.4.8Diagonal Matrix

In a diagonal matrix, all the elements above and below the diagonal

are zeros.


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2.4.9Scalar Matrix

A scalar matrix is a diagonal matrix in which the diagonal elements

are equal.

2.4.10Identity Matrix

An identity matrix is a diagonal matrix in which the diagonal

elements are equal to 1.

2.4.11Transpose Matrix

Given matrix A, the transpose of matrix A is another matrix where

the elements in the columns and rows have switched. In other words, the

rows become the columns and the columns become the rows.

(At)t= A

(A + B)t= At+ Bt

(A)t= A t

(A B)t= Bt At

2.4.12Regular Matrix

A regular matrix is a square matrix that has an inverse.


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2.4.13Singular Matrix

A singular matrix is a square matrix that has no inverse.

2.4.14Idempotent Matrix

The matrix A is idempotent if:

A2= A.

2.4.15Involutive Matrix

The matrix A is involutive if:

A2= I.

2.4.16Symmetric Matrix

A symmetric matrix is a square matrix that verifies:

A = At.

2.4.17Antisymmetric Matrix

An antisymmetric matrix is a square matrix that verifies:

A = A t.

2.4.18Orthogonal Matrix

A matrix is orthogonal if it verifies that:

A At = I.


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CHAPTERR 3

Analysis and Discussion


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3.1. Properties of matrix operation

3. 1.1 Properties of Addition

The basic properties of addition for real numbers also hold true for matrices.

Let A, B and C be m x n matrices

1. A + B = B + A commutative

2. A + (B + C) = (A + B) + C associative

3. There is a unique m x n matrix O with

A + O = A additive identity

4. For any m x n matrix A there is an m x n matrix B (called -A) with

A + B = O additive inverse

3.1.2 Properties of subtraction

Two matrices may be subtracted only if they have the same dimension; that is, they must have

the same number of rows and columns.Subtraction is accomplished by subtracting corresponding elements. For example, consider

matrix A and matrix B.

1 2 3 5 6 7

A = B= =

7 8 9 3 4 5

Both matrices have the same number of rows and columns(2 rows and 3 columns), so they can

be subtracted

1-5 2-6 3-7 -4 -4 -4

A-B = =

7-3 8-4 9-5 4 4 4


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3.1.3 Properties of Matrix Multiplication

Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to

matrices. Matrices rarely commute even if AB and BA are both defined. There often is no

multiplicative inverse of a matrix, even if the matrix is a square matrix. There are a few

properties of multiplication of real numbers that generalize to matrices. We state them now.

Let A, B and C be matrices of dimensions such that the following are defined. Then

1. A(BC) = (AB)C associative

2. A(B + C) = AB + AC distributive

3. (A + B)C = AC + BC distributive

4. There are unique matrices Imand Inwith

ImA = A In = A multiplicative identity

We will often omit the subscript and write I for the identity matrix. The identity matrix is a

square scalar matrix with 1's along the diagonal. For example

We will prove the second property and leave the rest for you.

3.1.4 Properties of Scalar Multiplication

Since we can multiply a matrix by a scalar, we can investigate the properties that this

multiplication has. All of the properties of multiplication of real numbers generalize. In

particular, we have

Let r and s be real numbers and A and B be matrices. Then

1. r(sA) = (rs)A

2. (r + s)A = rA + sA

3. r(A + B) = rA + rB

4. A(rB) = r(AB) = (rA)B


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3.1.5 Properties of the Transpose of a Matrix

Recall that the transpose of a matrix is the operation of switching rows and columns. We state

the following properties. We proved the first property in the last section.

Let r be a real number and A and B be matrices. Then

1. (AT)T = A

2. (A + B)T = AT+ BT

3. (AB)T = BTAT

4. (rA)T = rAT


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3.2 Matrix Operation

3.2.1 Matrix Equality

For two matrices to be equal, they must have

1. The same dimensions.-Each matrix has the same number of rows

-Each matrix has the same number of columns

2. Corresponding elements must be equal.

In other words, say that An x m= [aij] and that Bp x q= [bij].

Then A= Bif and only if n=p, m=q, and aij=bijfor all i and j in range.

Here are two matrices which are not equal even though they have the same elements.

Consider the three matrices shown below.

If A = B then we know that x = 34 and y = 54, since corresponding elements of equal

matrices are also equal.We know that matrix C is not equal to A or B, because C has more columns.

Note:

Two equal matrices are exactly the same.

If rows are changed into columns and columns into rows, we get a transpose matrix. If

the original matrix is A, its transpose is usually denoted by A' or At.

If two matrices are of the same order (no condition on elements) they are said to be

comparable.

If the given matrix A is of the order m x n, then its transpose will be of the order n x m.

Example 1:

The notation below describes two matricesAandB.

where i= 1, 2, 3 and j = 1, 2


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3.2.2 Matrix Addition

If two matrices have the same number of rows and same number of columns, then

the matrix sum can be computed:

If Ais an MxN matrix, and Bis also an MxN matrix, then their sum is an MxN matrix

formed by adding corresponding elements of Aand B

Here is an example of this:

Of course, in most practical situations the elements of the matrices are real numbers with

decimal fractions, not the small integers often used in examples.

3.2.3 Matrix Subtraction

If Aand Bhave the same number of rows and columns, then A- Bis defined as A+ (-

B). Usually you think of this as:

To form A- B, from each element of Asubtract the corresponding element of B.

Here is a partly finished example:

Notice in particular the elements in the first row of the answer. The way the result was

calculated for the elements in row 1 column 2 is sometimes confusing.

3.2.4 Matrix Multiplication

How to multiply two matrices

Matrix multiplication falls into two general categories:

Scalar in which a single number is multiplied with everyentry of a matrix

Multiplication of an entire matrix by another entire matrix For the rest of thepage, matrix multiplicationwill refer to this second category.
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3.2.5 Scalar Matrix Multiplication

In the scalar variety, everyentry is multiplied by a number, called ascalar.

What is the answer to the scalar multiplication problem below?

(See how this problem can be represented as a Scalar Dilation)

What is matrix multiplication?

Answer:You can multiply two matrices if, and only if, the number ofco lumnsinthe first matrix equals the number ofrowsin the second matrix.

Otherwise, the product of two matrices is undefined.

Theproduc tmatrix'sdimens ionsare

(rows of first matrix) (columns of the second matrix )
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In the picture on the

left, the matrices can be

multiplied since the

number ofcolumnsin

the 1st one, matrix A,

equals the number

ofrowsin the 2nd,

matrix B.

TheDimensionsof the product matrix

Rowsof 1stmatrix Columnsof 2

nd

Example 1

If we multiply a 23 matrix with a 31 matrix, the product matrix is 21

Here is how we get M11and M12in the product.

M11= r11 t11 + r12 t21 + r13t31M12= r21 t11 + r22 t21 + r23t31

Two Matrices that cannotbe multipliedMatrix C and D belowcannotbe multiplied together because the number of columns inCdoes not equalthe number of rows in D. In this case, the multiplication of these two
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matrices is not defined.

Practice Problem

Is the product of matrix A and Matrix B below defined orundef ined?

Answer
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Ok, so how do we multiply two matrices?

In order to multiply matrices,

Step 1: Make sure that the the number ofcolumnsin the 1st

one equals thenumber ofrowsin the 2ndone.(The pre-requisite to be able to multiply)

Step 2: Multiply the elements of each row of the first matrix by the elements ofeach column in the second matrix.

Step 3: Add the products.It's easier to understand if you go through the power point examples below.

Multiplication of Vectors

This combination of words "multiplication" and "vector" appears in at least four circumstances:

1. multiplication of a vector by a scalar2. scalar multiplication of vectors

3. multiplication of a vector by a matrix

4. vector multiplication of vectors

of which only the fourth may be looked at as a (semi)groupoperation. Although the restare also important, here I'll discuss only the latter. The vector multiplication (product)isdefined for 3-dimensional vectors. To proceed, we need the notion of right-and left-handednesswhich apply to three mutually perpendicular vectors.

Two noncollinear(non-parallel) vectors define a plane, and

there are two ways to erect a third vector perpendicular to thatplane (and, hence, to the two given vectors.) They are

distinguished by the right-or left-handed rules. The direction

defined by the right-handed rule is customarily preferred to

the other one. When one looks from the top of the forefinger

(z) the motion from the middle finger (x) towards the thump (y) is positive

(counterclockwise).

The late Isaac Isimov once suggested apprehensively that technological advances may

lead to thesaurus changes that would eliminate such dear to the heart notions as

the clockwiseandcounterclockwisedirections. Luckily, no technological progress could

possibly affect the physical underpinning of the right-handed rule.
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Definition

Let aand bbe two noncollinearvectors. Their cross(or external, or vector)productis

defined as a vector abperpendicular to both a and b and whose

direction is such that the three vectors a, b, and abform a right-handed

system.

length equals the area of the parallelogram built on the vectors aand b.

Cross product of collinear vectors is defined as 0. (Which is consistent with the

noncollinear case as we may think of two parallel vectors as defining a (one line)

parallelogram with zero area.

Obviously the product has no unit element. One the positive

side, both the associative and distributive laws hold. For the

latter, it's obvious from the geometric considerations. The

distributive lawimplies homogeneity(provided, of course, we

first establish some kind of continuity. But this is feasible:

small changes in eitheraorbresult in small changes of the

area of the parallelogram they define. The plane does not

change drastically either.) For a scalar t,

(ta)b=a(tb) = t(ab)

The cross product is alsoanticommutative:

ab= -ba

as it follows from the definition.

The cross product can be expressed in terms of a 33 determinant. Let e1, e2, and e3be

three mutually orthogonal unit vectors that form a right-handed system. Then, again by

definition,

e3= e1e2, e2= e3e1, e1= e2e3

If a= a1e1+ a2e2+ a3e3and b= b1e1+ b2e2+ b3e3then
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ab= (a2b3- a3b2)e1- (a1b3- a3b1)e2+ (a1b2- a2b1)e3

which is often written as

Theassociative law does not hold for the cross product. I.e., in general,

a(bc) (ab)c.

For example,

e1(e1e2) =e1e3= -e2

whereas

(e1e1)e2= 0e2= 0.

To compensate, there are other useful properties, e.g.,

a(bc) = (ac)b- (ab)c.

Especially useful is the mixed product of three vectors:

a(bc) = det(abc),

where the dot denotes thescalar product and the determinant det(abc) has

vectorsa,b,cas its columns. The determinant equals the volume of the parallelepiped

formed by the three vectors.

3.2.6 Matrix Inverse

The inverse of asquare matrix , sometimes called a reciprocal matrix, is a matrix

such that

(1)

where is theidentity matrix.Courant and Hilbert (1989, p. 10) use the notation to

denote the inverse matrix.

Asquare matrix has an inverseiff thedeterminant (Lipschutz 1991, p. 45). Amatrix possessing an inverse is callednonsingular,or invertible.
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The matrix inverse of asquare matrix may be taken inMathematicalusing thefunctionInverse[m].

For a matrix

(2)

the matrix inverse is

(3)

(4)

For a matrix

(5)

the matrix inverse is

(6)

A general matrix can be inverted using methods such as theGauss-Jordanelimination,Gaussian elimination,orLU decomposition.

The inverse of aproduct ofmatrices and can be expressed in terms of

and . Let

(7)

Then

(8)

and

(9)

Therefore,
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(10)

so

(11)

where is theidentity matrix,and

3.3 Elementary Matrix Operations

Elementary matrix operations play an important role in many matrix algebra

applications, such asfinding the inverse of a matrix andsolving simultaneous linear

equations.

3.3.1 Elementary Operations

There are three kinds of elementary matrix operations.

1. Interchange two rows (or columns).

2. Multiply each element in a row (or column) by a non-zero number.

3. Multiply a row (or column) by a non-zero number and add the result to another

row (or column).

When these operations are performed on rows, they are called elementary row

operations; and when they are performed on columns, they are called elementary

column operations.

3.3.2 Elementary Operation Notation

In many references, you will encounter a compact notation to describe elementary

operations. That notation is shown below.

Operation description Notation

Row

operations

1. Interchange rows iandj RIRj

2. Multiply row iby s, where s 0 sRi-->Ri
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3. Add s times row i to row j sRi+ Rj-->Rj

Column

operations

1. Interchange columns i and j CiCj

2. Multiply column i by s, where s 0 sCi-->Ci

3. Add s times column i to column j sCi+ Cj-->Cj


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CHAPTER- 4

Applications of Matrices


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4.1 Application of Matrix

There are numerous applications of matrices, both in mathematics and other sciences.

Some of them merely take advantage of the compact representation of a set of numbers

in a matrix. For example, in game theory and economics, the payoff matrix encodes thepayoff for two players, depending on which out of a given (finite) set of alternatives the

players choose.

4.1.1 Solving Linear Equations

Using matrix methods we can represent a system of linear equations and solve the

equations efficiently. Suppose we have a system of equations

This set of equations can be expressed compactly as augmented matrix form as follows

The row operations shown in chapter three perform the basic steps we used to solve

systems using elimination on an augmented matrix. This enables us to focus on the

numberswithout being concerned about algebraic manipulations.

This can be also solved by other method of Matrices. These are more easier to solve than

algebraic manipulations.

4.1.2 Electronics

Traditional mesh analysis in electronics leads to a system of linear equations that can be

described with a matrix. The behavior of many electronic components can be described

using matrices. Let A be a 2-dimensional vector with the component's input voltage v1and input current i1 as its elements, and let B be a 2-dimensional vector with the


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component's output voltage v2 and output current i2 as its elements. Then the behavior of

the electronic component 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.

4.1.3 Symmetries and transformations in physics

Physicists use a convenient matrix representation known as the Gell-Mann matrices,

which are used for the special unitary group SU gauge group that forms the basis of the

modern description of strong nuclear interactions, quantum chromo dynamics. The

CabibboKobayashiMaskawa matrix, in turn, expresses the fact that the basic quark

states that are important for weak interactions are not the same as, but linearly related to

the basic quark states that define particles with specific and distinct masses.

4.1.4Analysis and geometry

Geometrical optics provides further matrix applications. In this approximative theory, the

wave nature of light is neglected. The result is a model in which light rays are indeed

geometrical rays. If the deflection of light rays by optical elements is small, the action of

a lens or reflective element on a given light ray can be expressed as multiplication of a

two-component vector with a two-by-two matrix called ray transfer matrix: the vector's

components are the light ray's slope and its distance from the optical axis, while thematrix encodes the properties of the optical element. Actually, there are two kinds of

matrices, viz. a refraction matrix describing the refraction at a lens surface, and a

translation matrix, describing the translation of the plane of reference to the next

refracting surface, where another refraction matrix applies. The optical system,

consisting of a combination of lenses and/or reflective elements, is simply described by

the matrix resulting from the product of the components' matrices.

4.1.5Probability theory and statistics

Stochastic matrices are square matrices whose rows areprobability vectors, i.e., whose

entries are non-negative and sum up to one. Stochastic matrices are used to

defineMarkov chains with finitely many states. A row of the stochastic matrix gives the

probability distribution for the next position of some particle currently in the state that

corresponds to the row. Properties of the Markov chain like absorbing states,i.e., states

that any particle attains eventually, can be read off the eigenvectors of the transition

matrices.

Statistics also makes use of matrices in many different forms.Descriptive statistics is

concerned with describing data sets, which can often be represented asdata matrices,which may then be subjected todimensionality reductiontechniques. Thecovariance
http://en.wikipedia.org/wiki/Stochastic_matrixhttp://en.wikipedia.org/wiki/Probability_vectorhttp://en.wikipedia.org/wiki/Markov_chainhttp://en.wikipedia.org/wiki/Absorbing_statehttp://en.wikipedia.org/wiki/Descriptive_statisticshttp://en.wikipedia.org/wiki/Data_matrix_(multivariate_statistics)http://en.wikipedia.org/wiki/Dimensionality_reductionhttp://en.wikipedia.org/wiki/Covariance_matrixhttp://en.wikipedia.org/wiki/Covariance_matrixhttp://en.wikipedia.org/wiki/Dimensionality_reductionhttp://en.wikipedia.org/wiki/Data_matrix_(multivariate_statistics)http://en.wikipedia.org/wiki/Descriptive_statisticshttp://en.wikipedia.org/wiki/Absorbing_statehttp://en.wikipedia.org/wiki/Markov_chainhttp://en.wikipedia.org/wiki/Probability_vectorhttp://en.wikipedia.org/wiki/Stochastic_matrix


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matrix encodes the mutualvariance of severalrandom variables.Another technique

using matrices arelinear least squares,a method that approximates a finite set of pairs

(x1,y1), (x2,y2), ..., (xN,yN), by a linear function

yiaxi+ b, i= 1, ...,N

It can be formulated in terms of matrices, related to the singular value decomposition of matrices.

Random matrices are matrices whose entries are random numbers, subject to

suitableprobability distributions, such asmatrix normal distribution. Beyond probability theory,

they are applied in domains ranging fromnumber theory tophysics.

4.1.6 Cryptography

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. 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.

4.2. Application of Matrices in Real Life

Matrices find many applications in scientific fields and apply to practical real life

problems as well, thus making an indispensable concept for solving many practical

problems.

Some of the main applications of matrices are briefed below:

In physics related applications, matrices are applied in the study of electrical

circuits, quantum mechanics and optics. In geology, matrices are used for taking seismic surveys. They are used for

plotting graphs, statistics and also to do scientific studies in almost different

fields.

Matrices are used in representing the real world datas like the traits of peoples

population, habits, etc. They are best representation methods for plotting the

common survey things.

In computer based applications, matrices play a vital role in the projection of

three dimensional image into a two dimensional screen, creating the realistic

seeming motions.

The matrix calculus is used in the generalization of analytical notions like

exponentials and derivatives to their higher dimensions.
http://en.wikipedia.org/wiki/Covariance_matrixhttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)http://en.wikipedia.org/wiki/Singular_value_decompositionhttp://en.wikipedia.org/wiki/Random_matrixhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Matrix_normal_distributionhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Matrix_normal_distributionhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Random_matrixhttp://en.wikipedia.org/wiki/Singular_value_decompositionhttp://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)http://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Covariance_matrix


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In the calculation of battery power outputs, resistor conversion of electrical

energy into another useful energy, these matrices play a major role in

calculations. Especially in solving the problems using Kirchoffs laws of voltage

and current, the matrices are essential.

For Search Engine Optimization (SEO) 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.

One of the most important usages of matrices in computer side applications are

encryption of message codes. Matrices and their inverse matrices are used for a

programmer for coding or encrypting a message.

With the help of matrices internet functions are working and even banks could

work with transmission of sensitive and private datas.

In robotics and automation, matrices are the base elements for the robot

movements. The movements of the robots are programmed with the calculation

of matrices rows and columns. The inputs for controlling robots are given based

on the calculations from matrices.

Matrices are used in many organizations such as for scientists for recording the

data for their experiments.

Matrices are used in calculating the gross domestic products in economics whicheventually helps in calculating the goods production efficiently.


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CHAPTER- 5

Findings andRecommendation


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5.1 Findings

Matrices are one of the most powerful tools in Mathematics. The evolution of concept of

matrices is the result of a n attempt to obtain compact and simple methods of solving

system of linear equation. Matrix notation and operations are used in electronic

spreadsheet, programs for personal computer which in turn is used in different areas of

business and science like budgeting, sales projection, cost estimation etc. Also many

physical operations such as magnifications, rotations and reflection through a plane can

be represented mathematically by matrices. This mathematical tool is not only used in

certain branches of sciences but also in genetics, economics, sociology, modern

psychology and in industrial management. Along with its immeasurable benefits it has

some limitation also. Some general limitations of matrices are the followings:

Complicated calculations.

Difficulty in finding DETERMINANT of a 4 * 4matrix and more.

Time consuming.

Inappropriate and doubtful results.

Lengthy procedure involved.

Tends to create confusion which increases theproportion of mistakes.

Problems with Gauss Elimination

Not quite as easy to remember the procedure for hand solutions. Round off error may become significant, but can be partially mitigated by using

more advanced techniques such as pivotingor scaling.

Problems with Cramers Rule

Taking a long time. For 8 equations 2540160 operations, or around 700 hours it

requires one operation per second.

Requires a Square system

Round off error may become significant on large problems with non-integer

coefficients.

Doesnt always work if determinant of the coefficient matrix is zero

5.2 Recommendation

A major branch of numerical analysis is devoted to the development of efficient

algorithms for matrix computations, a subject that is centuries old and is today an

expanding area of research. Matrix decomposition methods simplify computations, both


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theoretically and practically. Because of some drawbacks matrices are not frequently

used like other mathematical methods.

For more efficient and effective use, matrices should apply in all possible sectors and

should practice more and more.

To be more friendly with matrices and its application:

It can be used for computer programming.

Can be used in business for budgeting, sales projection and cost estimation

Scientist can use a spreadsheet to analyze the result of experience

Can be used to compute industry income tax.

Can be used to analysis production and labor cost in industry.

Can be used in allocation of resources and production scheduling.

5.2 Conclusion

There are numerous applications of matrices, both in mathematics and other sciences.

Some of them merely take advantage of the compact representation of a set of numbers

in a matrix. For example, in game theory and economics, the payoff matrix encodes the

payoff for two players, depending on which out of a given (finite) set of alternatives the

players choose. In addition theoretical knowledge of properties of matrices and their

relation to other fields, it is important for practical purposes to perform matrix

calculations effectively and precisely. Many problems can be solved by both direct

algorithms and iterative approaches. For example, finding eigenvectors can be done by

finding a sequence of vectors xnconverging to an eigenvector when n tends to infinity.

Even matrices are very ancient mathematical concept but it has many applications in our

life.

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Matrices and Its Application