APPLICATIONS OF ENGINEERING MATHEMATICS

MATRICES AND ITS APPLICATIONS

Santhosh Kumar .S,

Venkatesh .S,

KARPAGAM INSTITUTE OF TECHNOLOGY

COIMBATORE

DEPARTMENT OF MECHANICAL ENGINEERING

Abstract

Engineering Mathematics is applied in daily life all in known and unknown ways. Branch of engineering mathematics are vector algebra, differential calculus, integration, discrete mathematics, matrices and determinant, etc. Among various topic, Matrices is generally interesting. Matrices have a long history of application in solving linear equations. between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations, including the concept of determinants, Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices. The term "matrix” was coined by Sylvester, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows.

Here in this paper you will be clear about Matrices definition – types – applications of matrices – graph theory – secret writing – cryptography – types of cryptography– conclusion.

Definition:

A matrix is a rectangular arrangement of mathematical expressions that can be simply numbers. For example,

An alternative notation uses large parentheses instead of box brackets:

Basic operations:

Addition:

The sum A+B of two m-by-n matrices A and B is just like this example,

A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Scalar multiplication:

The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:

(cA)i,j = c · Ai,j.

Transpose:

The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:

(AT)i,j = Aj,i.

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B:

,

Where 1 ≤ i ≤ m and 1 ≤ j ≤ p.[5] For example, the underlined entry 1 in the product is calculated as (1 × 1) + (0 × 1) + (2 × 0) = 1:

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined.[6] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they need not be equal, i.e., generally one has

AB ≠ BA,

i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:

Whereas

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.

It is called identity matrix because multiplication with it leaves a matrix unchanged: MIn = ImM = M for any m-by-n matrix M.

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, hence arise in solving matrix equations such as the Sylvester equation.

A particular case of matrix multiplication is tightly linked to linear equations: if x designates a column vector (i.e., n×1-matrix) of n variables x1, x2... xn, and A is an m-by-n matrix, then the matrix equation

Ax = b,

Where b is some m×1-column vector, is equivalent to the system of linear equations

A1, 1x1 + A1,2x2 + ... + A1,nxn = b1

...

Am,1x1 + Am,2x2 + ... + Am,nxn = bm 

This way, matrices can be used to compactly write and deal with multiple linear equations, i.e., systems of linear equations.

Now let us about the various applications of Matrices that are applied interestingly.

Graph theory:

The adjacency matrix of a finite graph is a basic notion of graph theory.

Linear combinations of quantum states in Physics:

The first model of quantum mechanics by Heisenberg in 1925 represented the theory's operators by infinite-dimensional matrices acting on quantum states. This is also referred to as matrix mechanics.

Computer graphics:

4×4 transformation rotation matrices are commonly used in computer graphics.

Solving linear equations

Using Row reduction

Cramer's Rule (Determinants)

Using the inverse matrix

Cryptography.

Write, encode, decode and send secret messages using Matrices:

Word games and mathematical puzzles often center on codes and secret messages. But secret messages aren't just for fun and games; they're used all over the world, and in all kinds of circumstances. Governments and military organizations use them to keep secrets; websites use them to keep financial information like credit card numbers and bank account information secret. And everyone enjoys sharing secret messages with friends. There are all kinds of codes you can

use to communicate with friends. Some are very complex and difficult to decode, and others are very simple. Some use numbers and mathematics, and others use the alphabet, or pictures and symbols.Not all codes are designed to keep secrets, though. Can you think of a code which was designed to send messages by telegraph, using sequences of short and long tones called dots and dashes?

Another system of writing looks like a code, but in reality is designed to help people who cannot see. The dots that make up letters are raised from the page so the blind person can feel them with the fingertips. Do you know what that system of writing is called?

Here at The Problem Site's "Codes, Decoding, and Secret Messages" site, you can learn more about a lot of different codes, and even try them out! Just click on any of the mysterious symbols at the top of the page to learn more about a code.

The Matrix Code is a complex method for creating and decoding secret messages. I won't go into all the details here, because it is very confusing if you haven't learned about matrices and determinants in your math class. And if you aren't in high school or college yet, you probably haven't! Most of the codes you've looked at her change the message one letter at a time. First, letter number one gets changed (into a number, a symbol, or another letter), then the second letter, and the third, and so on. But in a matrix code, the letters get changed in groups! So it's much harder to decode the message. 

This way of sharing secret messages are communicated by the following methods;

Steganography Cryptography

Steganography

There are a large number of steganographic methods that most of us are familiar with, ranging from invisible ink and microdots to secreting a hidden message in the second letter of each word of a large body of text and spread spectrum radio communication. With computers and networks, there are many other ways of hiding information, such as:

Covert channels Hidden text within Web pages Hiding files in "plain sight" Null ciphers Steganography

Today, however, is significantly more sophisticated than the examples above suggest, allowing a user to hide large amounts of information within image and audio files. These forms of steganography often are used in conjunction with cryptography so that the information is doubly protected; first it is encrypted and then hidden so that an adversary has to first find the information and then decrypt it.

There are a number of uses for steganography besides the mere novelty. One of the most widely used applications is for so-called digital watermarking. A watermark, historically, is the replication of an image, logo, or text on paper stock so that the source of the document can be at least partially authenticated. A digital watermark can accomplish the same function; a graphic artist, for example, might post sample images on her Web site complete with an embedded signature so that she can later prove her ownership in case others attempt to portray her work as their own.

Stego can also be used to allow communication within an underground community. There are several reports, for example, of persecuted religious minorities using steganography to embed messages for the group within images that are posted to known Web sites.

STEGANOGRAPHIC METHODS

The following formula provides a very generic description of the pieces of the steganographic process:

Cover medium + hidden data + Stego key = Stego medium

In this context, the cover medium is the file in which we will hide the hidden data, which may also be encrypted using the stego_key. The resultant file is the stego medium (which will, of course. be the same type of file as the cover_medium). The cover medium is typically image or audio files. In this article, I will focus on image files and will, therefore, refer to the cover image and stego image.

Before discussing how information is hidden in an image file, it is worth a fast review of how images are stored in the first place. An image file is merely a binary file containing a binary representation of the color or light intensity of each picture element (pixel) comprising the image.

Images typically use either 8-bit or 24-bit color. When using 8-bit color, there is a definition of up to 256 colors forming a palette for this image, each color denoted by an 8-bit value. A 24-bit color scheme, as the term suggests, uses 24 bits per pixel and provides a much better set of colors. In this case, each pix is represented by three bytes, each byte representing the intensity of the three primary colors red, green, and blue (RGB), respectively. The Hypertext Markup Language (HTML) format for indicating colors in a Web page often uses a 24-bit format employing six hexadecimal digits, each pair representing the amount of red, blue, and green, respectively.

The size of an image file, then, is directly related to the number of pixels and the granularity of the color definition. A typical 640x480 pix image using a palette of 256 colors would require a file about 307 KB in size (640 • 480 bytes), whereas a 1024x768 pix high-resolution 24-bit color image would result in a 2.36 MB file (1024 • 768 • 3 bytes).

To avoid sending files of this enormous size, a number of compression schemes have been developed over time, notably Bitmap (BMP), Graphic Interchange Format (GIF), and Joint

Photographic Experts Group (JPEG) file types. Not all are equally suited to steganography, however.

GIF and 8-bit BMP files employ what is known as lossless compression, a scheme that allows the software to exactly reconstruct the original image. JPEG, on the other hand, uses lossy compression, which means that the expanded image is very nearly the same as the original but not an exact duplicate. While both methods allow computers to save storage space, lossless compression is much better suited to applications where the integrity of the original information must be maintained, such as steganography. While JPEG can be used for stego applications, it is more common to embed data in GIF or BMP files.

The simplest approach to hiding data within an image file is called least significant bit (LSB) insertion. In this method, we can take the binary representation of the hidden data and overwrite the LSB of each byte within the cover image. If we are using 24-bit color, the amount of change will be minimal and indiscernible to the human eye. As an example, suppose that we have three adjacent pixels (nine bytes) with the following RGB encoding:

10010101 00001101 1100100110010110 00001111 1100101010011111 00010000 11001011

Now suppose we want to "hide" the following 9 bits of data (the hidden data is usually compressed prior to being hidden): 101101101. If we overlay these 9 bits over the LSB of the 9 bytes above, we get the following (where bits in bold have been changed):

10010101 00001100 1100100110010111 00001110 1100101110011111 00010000 11001011

Note that we have successfully hidden 9 bits but at a cost of only changing 4, or roughly 50%, of the LSBs.

This description is meant only as a high-level overview. Similar methods can be applied to 8-bit color but the changes, as the reader might imagine, are more dramatic. Gray-scale images, too, are very useful for steganographic purposes. One potential problem with any of these methods is that they can be found by an adversary who is looking. In addition, there are other methods besides LSB insertion with which to insert hidden information.

Without going into any detail, it is worth mentioning steganalysis, the art of detecting and breaking steganography. One form of this analysis is to examine the color palette of a graphical image. In most images, there will be a unique binary encoding of each individual color. If the image contains hidden data, however, many colors in the palette will have duplicate binary encodings since, for all practical purposes, we can't count the LSB. If the analysis of the color

palette of a given file yields many duplicates, we might safely conclude that the file has hidden information.

But what files would you analyze? Suppose I decide to post a hidden message by hiding it in an image file that I post at an auction site on the Internet. The item I am auctioning is real so a lot of people may access the site and download the file; only a few people know that the image has special information that only they can read. And we haven't even discussed hidden data inside audio files. Indeed, the quantity of potential cover files makes steganalysis a Herculean task.

Network steganography covers a broad spectrum of techniques, which include, among others:

Steganophony - the concealment of messages in voice over in conversations, e.g. the employment of delayed or corrupted packets that would normally be ignored by the receiver (this method is called LACK - Lost Audio Packets Steganography), or, alternatively, hiding information in unused header fields.

WLAN Steganography – the utilization of methods that may be exercised to transmit steganograms in Wireless Local Area Networks. A practical example of WLAN Steganography is the HICCUPS system (Hidden Communication System for Corrupted Networks).

Network steganography:It is a modern version of an old idea. With today's technology, information can be smuggled in essentially any type of digital file, including JPEGs or bitmaps, MP3s or WAV files, and MPEG movies. More than a hundred such steganographic applications are freely available on the Internet. Many of these programs are slick packages whose use requires no significant technical skills whatsoever. Typically, one mouse click selects the carrier, a second selects the secret information to be sent, and a third sends the message and its secret cargo. All the recipient needs is the same program the sender used; it typically extracts the hidden information within seconds.Any binary file can be concealed—for instance, pictures in unusual formats, software (a nasty virus, say), or blueprints. The favored carrier files are the most common ones, like JPEGs or MP3s. This emphasis on popular file formats increases the anonymity of the entire transaction, because these file types are so commonplace that they don't stick out.The one limitation that steganographers have traditionally faced is file size. The rule of thumb is that you can use 10 percent of a carrier file's size to smuggle data. For an ambitious steganographer, that could be a problem: Imagine an electronic equipment factory employee trying to explain to the IT department why he has to send his mother a 100-megabyte picture of the family dog. For that reason, steganographers soon turned to audio and video files. A single 6-minute song, in the MP3 compression format, occupies 30 MB; it's enough to conceal every play Shakespeare ever wrote.And yet, even with these precautions, conventional steganography still has an Achilles' heel: It leaves a trail. Pictures and other e-mail attachments stored on a company's outgoing e-mail servers retain the offending document. Anything sent has to bounce through some kind of relay and can therefore be captured, in theory.Steganography poses serious threats to network security mainly by enabling confidential information leakage. The new crop of programs leaves almost no trail. Because they do not hide information inside digital files, instead using the protocol itself, detecting their existence is nearly impossible.

All the new methods manipulate the Internet Protocol (IP), which is a fundamental part of any communication, voice or text based, that takes place on the Internet. The IP specifies how information travels through a network. Like postal service address standards, IP is mainly in charge of making sure that sender and destination addresses are valid, that parcels reach their destinations, and that those parcels conform to certain guidelines. All traffic, be it e-mail or streaming video, travels via a method called packet switching, which parcels out digital data into small chunks, or packets, and sends them over a network shared by countless users. IP also contains the standards for packaging those packets.Let's say you're sending an e-mail. After you hit the Send button, the packets travel easily through the network, from router to router, to the recipient's in-box. Once these packets reach the recipient, they are reconstituted into the full e-mail.The important thing is that the packets don't need to reach their destination in any particular order. IP is a "connectionless protocol," which means that one node is free to send packets to another without setting up a prior connection, or circuit. This is a departure from previous methods, such as making a phone call in a public switched telephone network, which first requires synchronization between the two communicating nodes to set up a dedicated and exclusive circuit. Within reason, it doesn't matter when packets arrive or whether they arrive in order.As you can imagine, this method works better for order-insensitive data like e-mail and static Web pages than it does for voice and video data. Whereas the quality of an e-mail message is immune to traffic obstructions, a network delay of even 20 milliseconds can very much degrade a second or two of video.To cope with this challenge, network specialists came up with the Voice over Internet Protocol (VoIP). It governs the way voice data is broken up for transmission the same way IP manages messages that are less time sensitive. VoIP enables data packets representing a voice call to be split up and routed over the Internet.The connection of a VoIP call consists of two phases: the signaling phase, followed by the voice-transport phase. The first phase establishes how the call will be encoded between the sending and receiving computers. During the second phase, data are sent in both directions in streams of packets. Each packet, which covers about 20 milliseconds of conversation, usually contains 20 to 160 bytes of voice data. The connection typically conveys between 20 and 50 such packets per second.Telephone calls must occur in real time, and significant data delays would make for an awkward conversation. So to ferry a telephone call over the Internet, which was not originally intended for voice communications, VoIP makes use of two more communications protocols, which had to be layered on top of IP: The Real-Time Transport Protocol (RTP) and the User Datagram Protocol (UDP). The RTP gets time-sensitive video and audio data to its destination fast and so has been heavily adopted in much of streaming media, such as telephony, video teleconference applications, and Web-based push-to-talk features. To do that, it relies in turn on the UDP.Because voice traffic is so time critical, UDP does not bother to check whether the data are reliable, intact, or even in order. So in a VoIP call, packets are sometimes stuck in out of sequence. But that's not a big deal because the occasional misplaced packet won't significantly affect the quality of the phone call. The upshot of UDP is that the protocol opens a direct connection between computers with no mediation, harking back to the era of circuit switching: Applications can send data packets to other computers on a connection without previously setting up any special transmission channels or data paths. That means it's completely private.

Compared to old-fashioned telephony, IP is unreliable. That unreliability may result in several classes of error, including data corruption and lost data packets. Steganography exploits those errors.Because these secret data packets, or "steganograms," are interspersed among many IP packets and don't linger anywhere except in the recipient's computer, there is no easy way for an investigator—who could download a suspect image or analyze an audio file at his convenience—to detect them.Recent applications of stenography:

Mobile phone and Internet technologies have progressed along each other. The importance of both these technologies has resulted in the creation of a new technology for establishing wireless Internet connection through mobile phone, known as Wireless Application Protocol (WAP). However, considering the importance of the issue of data security and especially establishing hidden communications, many methods have been presented. In the meanwhile, steganography is a relatively new method.In this paper, a method for hidden exchange of data has been presented by using steganography on WML pages (WML stands for Wireless Markup Language, which is a language for creating web pages for the WAP). The main idea in this method is hiding encoded data in the ID attribute of WML document tags. The coder program in this method has been implemented using the Java language. The decoder program to be implemented on the mobile phone has been written with a version of Java language specifically used for small devices, which is called J2ME (Java 2 Micro Edition). It was tested on a Nokia series 60 mobile phone.

What does cryptography mean?

• Cryptography is the science of information security.

• The word is derived from the Greek kryptos, meaning hidden.

• Cryptography includes techniques such as merging words with images, and other ways to hide information in storage or transit.

Cryptography involves encrypting data so that a third party cannot intercept and read the data.

In the early days of satellite television, the video signals weren't encrypted and anyone with a satellite dish could watch whatever was being shown. Well, this didn't work because all of the networks using satellites didn't want the satellite dish owners to be able to receive their satellite feed for no cost while cable subscribers had to pay for the channel, they were losing money. So, they started encrypting the video signal with a system called Videocipher.

What the Videocipher encryption system did was to convert the signal into digital form, encrypt it, and send the data over the satellite. If the satellite dish owner had a Videocipher box, and paid for the channel, then the box would descramble (unencrypted) the signal and return it to its original, useful form.

This was done by using a key that was invertible. It was very important that they key be invertible, or there would be no way to return the encrypted data to its original form.

The same thing can be done using matrices.

Encryption Process

1. Convert the text of the message into a stream of numerical values.2. Place the data into a matrix.3. Multiply the data by the encoding matrix.4. Convert the matrix into a stream of numerical values that contains the encrypted message.

Example

Consider the message "Red Rum"

A message is converted into numeric form according 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.

This data was placed into matrix form. The size of the matrix depends on the size of the encryption key. Let's say that our encryption matrix (encoding matrix) is a 2x2 matrix. Since I have seven pieces of data, I would place that into a 4x2 matrix and fill the last spot with a space to make the matrix complete. Let's call the original, unencrypted data matrix A.

    18 5  

A =

  4 0  

  18 21  

    13 0  

There is an invertible matrix which is called the encryption matrix or the encoding matrix. We'll call it matrix B. Since this matrix needs to be invertible, it must be square.

This could really be anything; it's up to the person encrypting the matrix. I'll use this matrix.

B =

  4 -2  

  -1

3  

The unencrypted data is then multiplied by our encoding matrix. The result of this multiplication is the matrix containing the encrypted data. We'll call it matrix X.

    67

-21  

X = A B =

  16

-8  

  51

27  

    52

-26  

The message that you would pass on to the other person is the the stream of numbers 67, -21, 16, -8, 51, 27, 52, -26.

Decryption Process

1. Place the encrypted stream of numbers that represents an encrypted message into a matrix.

2. Multiply by the decoding matrix. The decoding matrix is the inverse of the encoding matrix.

3. Convert the matrix into a stream of numbers.4. Convert the numbers into the text of the original message.

Example

The message you need to decipher is in the encrypted data stream 67, -21, 16, -8, 51, 27, 52, -26.

The encryption matrix is not transmitted. It is known by the receiving party so that they can decrypt the message. Other times, the inverse is known by the receiving party. The encryption matrix cannot be sent with the data, otherwise anyone could grab the data and decode the

information. Also, by not having the decoding matrix, someone intercepting the message doesn't know what size of matrix to use.

The receiving end gets the encrypted message and places it into matrix form.

    67 -21

 

X =  16 -8  

  51 27  

    52 -26

 

The receiver must calculate the inverse of the encryption matrix. This would be the decryption matrix or the decoding matrix.

B-1 =

  0.3 0.2

 

  0.1 0.4  

The receiver then multiplies the encrypted data by the inverse of the encryption matrix. The result is the original unencrypted matrix.

    18

5  

A = X B-1 =

  4 0  

  18

21  

    13

0  

The receiver then takes the matrix and breaks it apart into values 18, 5, 4, 0, 18, 21, 13, 0 and converts each of those into a character according to the numbering scheme. 18=R, 5=E, 4=D, 0=space, 18=R, 21=U, 13=M, 0=space.

Trailing spaces will be discarded and the message is received as intended: "RED RUM"

Applications of cryptography

Secrecy in Transmission: Most current secrecy systems for transmission use a private key system for transforming transmitted information because it is the fastest method that operates with reasonable assurance and low overhead. If the number of communicating parties is small, key distribution is done

periodically with a courier service and key maintenance is based on physical security of the keys over the period of use and destruction after new keys are distributed. If the number of parties is large, electronic key distribution is usually used. Historically, key distribution was done with a special key-distribution-key (also known as a master-key) maintained by all parties in secrecy over a longer period of time than the keys used for a particular transaction. The "session-key" is generated at random either by one of the parties or by a trusted third party and distributed using the master-key.

Secrecy in Storage: Secrecy in storage is usually maintained by a one-key system where the user provides the key to the computer at the beginning of a session, and the system then takes care of encryption and decryption throughout the course of normal use. As an example, many hardware devices are available for personal computers to automatically encrypt all information stored on disk. When the computer is turned on, the user must supply a key to the encryption hardware. The information cannot be read meaningfully without this key, so even if the disk is stolen, the information on it will not be useable. Secrecy in storage has its problems. If the user forgets a key, all of the information encrypted with it becomes permanently unusable. The information is only encrypted while in storage, not when in use by the user. This leaves a major hole for the attacker. If the encryption and decryption are done in software, or if the key is stored somewhere in the system, the system may be circumvented by an attacker. Backups of encrypted information are often stored in plaintext because the encryption mechanism is only applied to certain devices.

Authentication of Identity: Authenticating the identity of individuals or systems to each other has been a problem for a very long time. Simple passwords have been used for thousands of years to prove identity. More complex protocols such as sequences of keywords exchanged between sets of parties are often shown in the movies or on television. Cryptography is closely linked to the theory and practice of using passwords, and modern systems often use strong cryptographic transforms in conjunction with physical properties of individuals and shared secrets to provide highly reliable authentication of identity. Determining good passwords falls into the field known as key selection. In essence, a password can be thought of as a key to a cryptosystem that allows encryption and decryption of everything that the password allows access to. In fact, password systems have been implemented in exactly this way in some commercial products.

Credentialing Systems: A credential is typically a document that introduces one party to another by referencing a commonly known trusted party. For example, when credit is applied for, references are usually requested. The credit of the references is checked and they are contacted to determine the creditworthiness of the applicant. Credit cards are often used to credential an individual to attain further credit cards. A driver's license is a form of credential, as is a passport. Electronic credentials are designed to allow the credence of a claim to be verified electronically. Although no purely electronic credentialing systems are in widespread use at this time, many such systems are being integrated into the smart-card systems in widespread use in Europe. A smart-card is simply a credit-card shaped computer that performs cryptographic functions and

stores secret information. When used in conjunction with other devices and systems, it allows a wide variety of cryptographic applications to be performed with relative ease of use to the consumer.

Electronic Signatures: Electronic signatures, like their physical counterparts, are a means of providing a legally binding transaction between two or more parties. To be as useful as a physical signature, electronic signatures must be at least as hard to forge, at least as easy to use, and accepted in a court of law as binding upon all parties to the transaction.

Electronic Cash: There are patents under force throughout the world today to allow electronic information to replace cash money for financial transactions between individuals. Such a system involves using cryptography to keep the assets of nations in electronic form. Clearly the ability to forge such a system would allow national economies to be destroyed in an instant. The pressure for integrity in such a system is staggering.

Graph theory

Although a pictorial representation of a graph is very convenient for a visual study, other representations are better for computer processing. A matrix is a convenient and useful way of representing a graph to a computer. Matrices lend themselves easily to mechanical manipulations. Besides, many known results of matrix algebra can be readily applied to study the structural properties of graphs from an algebraic point of view. In many applications of graph theory, such as in electrical network analysis and operations research matrices also turn out to be the natural way of expressing the problem.

THEOREM 1 Two graphs G and G are isomorphic if and only if their incidence matrices A (G) and A (G) differ only by permutations of rows and columns.

THEOREM 2 If A (G) is an incidence matrix of a connected graph G with n vertices the rank of A (G) is n-1.

COROLLARY

The reduced incidence matrix of tree is nonsingular. A graph with n vertices and n-1 edges that is not a tree is disconnected. The rank of the incidence matrix of such a graph will be less than n-1.Therefore the (n-1) by (n-1) reduced incidence matrix of such a graph will not be nonsingular. In other words, the reduced incidence matrix of a graph is nonsingular if and only if the graph is tree.

THEOREM 3 Let A (G) be an incidence matrix of a connected graph G with n vertices. An (n-1)by (n-1) sub matrix of A(G) is nonsingular if and only if the n-1 edges corresponding to the n-1 columns of this matrix constitute a spanning tree in G.

THEOREM 4 Let A and B be, respectively, the circuit matrix and the incidence matrix (of a self – loop – free graph) whose columns are arranged using the same order of edges. Then every row of B is orthogonal to every row A;

That is, A.BT = B.AT = 0

CIRCUIT MATRIX

Let the number of different circuits in a graph G be q and the number of edges in G be e. Then a circuit matrix x B= [b] of jG is a q e, (0,1) matrix defined as follows.

b=1, if ith circuit includes jth edges and

=0 otherwise.

To emphasize the fact that B is a circuit matrix of graph the circuit matrix may also be written as B(G)

The graph in has four different circuits, {a,b},{c,e,g},{d,f,g} and {c,d f,e}.Therefore, its circuits matrix is a 4 by (0,1) matrix an shown.

[110 0 00 0000 1 010 1000 0 101 1000 1 111 00

]The following observation can be made about a circuit matrix B(G) of a graph G:

1. A column of all zeros corresponds to noncircuit edges.2. Each row of B(G) is a circuit vector.3. Unlike the incidence matrix a circuit matrix is capable of representing a self-loop the

corresponding row will have a single1.4. The number of 1’s in a row is equal to the number of edges in the corresponding circuit.5. If graph G is separable and consists of two blocks g1, g2, the matrix B(G) can be written

in a block diagonal form as

B (G) = [B (g 1) 00 B(g 2)],

Where B (g1) and B(g2) are the circuit matrices of g1and g2.This observation results from the fact that circuits in g1, have no edges belonging to g2, and vice versa.

6. Permutation of any two rows or columns in a circuit matrix simply corresponds to relabeling the circuits and edges.

Conclusion

Applications of Matrices are not only Graph theory, Stenography, cryptography. There are also many ideas applied in this field secret in banking, communication in military administration, confidential message transduction, computerized lockers, etc are other. Our scholars are still working in this field to develop a World Wide secured Communication for all people. Coding and Encoding a lot of message is ease when it combines with Software Development regarding. Hence Matrices is applied in many useful purposes in our World.