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APPLICATION

OF

MATRICES

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WHAT IS A MATRIX?

In mathematics, a matrix (pluralmatrices, orlesscommonlymatrixes) is a rectangulararray ofnumbers,such as

An item in a matrix is called an entry oran element.

Matrices of the same size can be added and subtractedentrywise and matrices of compatible sizes can bemultiplied. These operations have many of theproperties of ordinary arithmetic, except that matrixmultiplication is not commutative, that is, AB and BA

are not equal in general. Matrices consisting of only onecolumn orrow define the components ofvectors, whilehigher-dimensional(e.g., three-dimensional) arrays ofnumbers define the components of a generalization of avectorcalled a tensor.

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Applications of matricesFIELD OF APPLICATION OFMATRICES USAGE

COMPUTER DATA ENCRYPTION

ECONOMICS & COMMERCE CALCULATION OF GDP etc.

POLITICS CASTING POLLS

ENGINEERING MATH REPORT RECORDING

ARCHITECTURE USED ALONG WITH COMPUTING

SCIENTIFIC STUDY MAKING OF GRAPHS AND STATISTICS

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Applications of matrices

In

CRYPTOGRAPHY

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What is Cryptography?

Cryptography, to most people, is concernedwith keeping communications private.

Encryption is the transformation of data intosome unreadable form. Its purpose is toensure privacy by keeping the informationhidden from anyone for whom it is notintended, even those who can see the

encrypted data. Decryption is the reverse of encryption; it is

the transformation of encrypted data backinto some intelligible form.

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Why Cryptography? Cryptography involves encrypting data so that

a third party can not intercept and read the

data.

In the early days of satellite television, the

video signals weren't encrypted and anyonewith a satellite dish could watch whatever was

being shown.

This was a problem so data must be somehowhidden for this cryptography is used

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PROCESS OF ENCRYPTION:

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CRYPTOGRAPHY

SECRET EY

SYMMETRIC

ASYMMETRIC

PUBLIC EY OPEN ACCESS

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How does Encryption takes place?

Encryption Process:

Convert the text of the message into a stream

of numerical values. Place the data into a matrix.

Multiply the data by the encoding matrix.

Convert the matrix into a stream of numericalvalues that contains the encrypted message.

STEPS FOR

ENCRYPTION

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Consider the message PREPARE TO

NEGOTIATE"

A message is converted into numeric formaccording to some scheme. The easiest

scheme is to let space=0, A=1, B=2, ..., Y=25,

and Z=26. For example, the message "Red

Rum" would become 18, 5, 4, 0, 18, 21, 13.

STEP 1 :

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STEP 2:the encoding matrix be

We assign a number for each letter of thealphabet.

For simplicity, let us associate each letter with itsposition in the alphabet: A is 1, B is 2, and so on.

Also, we assign the number 27 (remember wehave only 26 letters in the alphabet) to a spacebetween two words. Thus the message becomes

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STEP 3

Since we are using a 3 by 3 matrix, we break the

enumerated message above into a sequence of 3 by 1

vectors:

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STEP 4:

Note that it was necessary to add a space at

the end of the message to complete the last

vector. We now encode the message by

multiplying each of the above vectors by the

encoding matrix. This can be done by writing

the above vectors as columns of a matrix and

perform the matrix multiplication of thatmatrix with the encoding matrix as follows:

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which gives the matrix

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STEP 5:

The columns of this matrix give the encoded

message. The message is transmitted in the

following linear form

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STEP 6 (DECODING):

To decode the message, the receiver writes thisstring as a sequence of 3 by 1 column matricesand repeats the technique using the inverse of

the encoding matrix.The inverse of this encoding matrix, the decoding

matrix, is:

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Thus, to decode the message, perform the

matrix multiplication

and get the matrix

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CONCLUDING STEP

The columns of this matrix, written in linear

form, give the original message

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FIBONACCI CRYPTOGRAPHY The general idea of the Fibonacci

cryptography is similar to the Fibonacci codingand based on the application of the

generalized Fibonacci matrices, the Qp-

matrices, for encryption and decryption of the

initial messageM.

Note that the encryption/decryption key is the pair of the numbers

ofp and n. Since p = 0, 1, 2, 3, ... and n = 1, 2, 3, ... this means that

this method has theoretically unlimited number of the

encryption/decryption keys.

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STEP 1:

Let's consider the Fibonacci encryption

method:

and then the Fibonacci decryption method

It follows from (1) and (2) that the Fibonacci encryption algorithm

(1) is reduced to the

n-

multiple multiplication of the initial matrix Mby the matrix Qpand the Fibonacci decryption algorithm is reduced to the n-

multiple multiplication of the secret message Eby the inverse

matrix .

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STEP 2: Let's consider now the multiplication of the

initial matrixM by the matrix Qp

. Let'sconsider the concrete example when theinitial message is represented in the form ofthe 4 X4 matrix:

For this case the Qp-matrix of the 4-d order (p +

1 = 4) is used for encryption:.1

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STEP 3: For calculation of the matrix ofE=M Q3 we

can represent it in the following form:

After the execution of the matrix multiplication (5) the matrix E

takes the following form:

.6

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CONCLUDING STEP

Comparing the initial matrix (3) with its secret equivalent(6) we can formulate the following rule concerning themultiplication of the initial matrix M by the coding Qp-mat.

RULE:

For the multiplication of the initial matrix (6) by the inversematrix it is necessary to shift all the matrix entries of theinitial matrix (6) to the left by one column, and form thelast entries of each row by means of the subtraction of thesecond entry of each row of the initial matrix from its first

entry.

Det E= Det M X (-1)pn. (9)

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CONCLUSION

THUS IN THIS WAY DATA CAN BE MADE SAFEAND MORE PROTECTED FROM THE VARIOUSHACKING DEVICES