Applications of Matrices


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Applications of Matrices

  2. 2. AbstractEngineering Mathematics is applied in daily life all in known and unknown ways. Branch ofengineering mathematics are vector algebra, differential calculus, integration, discretemathematics, matrices and determinant, etc. Among various topic, Matrices is generallyinteresting. Matrices have a long history of application in solving linear equations. between 300BC and AD 200, is the first example of the use of matrix methods to solve simultaneousequations, including the concept of determinants, Early matrix theory emphasized determinantsmore strongly than matrices and an independent matrix concept akin to the modern notionemerged only in 1858, with Cayleys Memoir on the theory of matrices. The term "matrix” wascoined by Sylvester, who understood a matrix as an object giving rise to a number ofdeterminants today called minors, that is to say, determinants of smaller matrices that derivefrom 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.
  3. 3. Definition:A matrix is a rectangular arrangement of mathematical expressions that can besimply 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.
  4. 4. Multiplication of two matrices is defined only if the number of columns of the left matrix is thesame 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 dotproduct of the corresponding row of A and the corresponding column of B: , [5]Where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example, the underlined entry 1 in the product is calculatedas (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 thesize of the matrices is such that the various products are defined.[6] The product AB may bedefined 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., generallyone 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: WhereasThe identity matrix In of size n is the n-by-n matrix in which all the elements on the maindiagonal are equal to 1 and all other elements are equal to 0, e.g.
  5. 5. It is called identity matrix because multiplication with it leaves a matrixunchanged: MIn = ImM = M for any m-by-n matrix M.Besides the ordinary matrix multiplication just described, there exist other less frequently usedoperations on matrices that can be considered forms of multiplication, hence arise in solvingmatrix equations such as the Sylvester equation.A particular case of matrix multiplication is tightly linked to linear equations: if x designates acolumn vector (i.e., n×1-matrix) of n variables x1, x2... xn, and A is an m-by-n matrix, then thematrix equationAx = 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 = bmThis 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 theorys 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 Cramers Rule (Determinants) Using the inverse matrix  Cryptography.
  6. 6. Write, encode, decode and send secret messages using Matrices:Word games and mathematical puzzles often center on codes and secret messages. But secretmessages arent just for fun and games; theyre used all over the world, and in all kinds ofcircumstances. Governments and military organizations use them to keep secrets; websites usethem 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 canuse to communicate with friends. Some are very complex and difficult to decode, and others arevery simple. Some use numbers and mathematics, and others use the alphabet, or pictures andsymbols.Not all codes are designed to keep secrets, though. Can you think of a code which was designedto 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 cannotsee. The dots that make up letters are raised from the page so the blind person can feel them withthe fingertips. Do you know what that system of writing is called?Here at The Problem Sites "Codes, Decoding, and Secret Messages" site, you can learn moreabout a lot of different codes, and even try them out! Just click on any of the mysterious symbolsat 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 wont go intoall the details here, because it is very confusing if you havent learned about matrices anddeterminants in your math class. And if you arent in high school or college yet, you probablyhavent! Most of the codes youve looked at her change the message one letter at a time. First, letternumber one gets changed (into a number, a symbol, or another letter), then the second letter, andthe third, and so on. But in a matrix code, the letters get changed in groups! So its much harderto decode the message.This way of sharing secret messages are communicated by the following methods; Steganography Cryptography SteganographyThere are a large number of steganographic methods that most of us are familiar with, rangingfrom invisible ink and microdots to secreting a hidden message in the second letter of each wordof 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
  7. 7. Today, however, is significantly more sophisticated than the examples above suggest, allowing auser to hide large amounts of information within image and audio files. These forms ofsteganography often are used in conjunction with cryptography so that the information is doublyprotected; first it is encrypted and then hidden so that an adversary has to first find theinformation and then decrypt it.There are a number of uses for steganography besides the mere novelty. One of the most widelyused applications is for so-called digital watermarking. A watermark, historically, is thereplication of an image, logo, or text on paper stock so that the source of the document can be atleast partially authenticated. A digital watermark can accomplish the same function; a graphicartist, for example, might post sample images on her Web site complete with an embeddedsignature so that she can later prove her ownership in case others attempt to portray her work astheir own.Stego can also be used to allow communication within an underground community. There areseveral reports, for example, of persecuted religious minorities using steganography to embedmessages for the group within images that are posted to known Web sites.STEGANOGRAPHIC METHODSThe following formula provides a very generic description of the pieces of the steganographicprocess: Cover medium + hidden data + Stego key = Stego mediumIn this context, the cover medium is the file in which we will hide the hidden data, which mayalso be encrypted using the stego_key. The resultant file is the stego medium (which will, ofcourse. be the same type of file as the cover_medium). The cover medium is typically image oraudio files. In this article, I will focus on image files and will, therefore, refer to the coverimage and stego image.Before discussing how information is hidden in an image file, it is worth a fast review of howimages are stored in the first place. An image file is merely a binary file containing a binaryrepresentation of the color or light intensity of each picture element (pixel) comprising theimage.Images typically use either 8-bit or 24-bit color. When using 8-bit color, there is a definition ofup to 256 colors forming a palette for this image, each color denoted by an 8-bit value. A 24-bitcolor scheme, as the term suggests, uses 24 bits per pixel and provides a much better set ofcolors. In this case, each pix is represented by three bytes, each byte representing the intensity ofthe three primary colors red, green, and blue (RGB), respectively. The Hypertext MarkupLanguage (HTML) format for indicating colors in a Web page often uses a 24-bit formatemploying six hexadecimal digits, each pair representing the amount of red, blue, and green,respectively.
  8. 8. The size of an image file, then, is directly related to the number of pixels and the granularity ofthe color definition. A typical 640x480 pix image using a palette of 256 colors would require afile about 307 KB in size (640 • 480 bytes), whereas a 1024x768 pix high-resolution 24-bit colorimage 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 beendeveloped over time, notably Bitmap (BMP), Graphic Interchange Format (GIF), and JointPhotographic 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 allowsthe 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 theoriginal 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 originalinformation must be maintained, such as steganography. While JPEG can be used for stegoapplications, 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 overwritethe LSB of each byte within the cover image. If we are using 24-bit color, the amount of changewill be minimal and indiscernible to the human eye. As an example, suppose that we have threeadjacent pixels (nine bytes) with the following RGB encoding: 10010101 00001101 11001001 10010110 00001111 11001010 10011111 00010000 11001011Now suppose we want to "hide" the following 9 bits of data (the hidden data is usuallycompressed prior to being hidden): 101101101. If we overlay these 9 bits over the LSB of the 9bytes above, we get the following (where bits in bold have been changed): 10010101 00001100 11001001 10010111 00001110 11001011 10011111 00010000 11001011Note that we have successfully hidden 9 bits but at a cost of only changing 4, or roughly 50%, ofthe LSBs.This description is meant only as a high-level overview. Similar methods can be applied to 8-bitcolor 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 isthat they can be found by an adversary who is looking. In addition, there are other methodsbesides LSB insertion with which to insert hidden information.
  9. 9. Without going into any detail, it is worth mentioning steganalysis, the art of detecting andbreaking steganography. One form of this analysis is to examine the color palette of a graphicalimage. In most images, there will be a unique binary encoding of each individual color. If theimage contains hidden data, however, many colors in the palette will have duplicate binaryencodings since, for all practical purposes, we cant count the LSB. If the analysis of the colorpalette of a given file yields many duplicates, we might safely conclude that the file has hiddeninformation.But what files would you analyze? Suppose I decide to post a hidden message by hiding it in animage file that I post at an auction site on the Internet. The item I am auctioning is real so a lot ofpeople may access the site and download the file; only a few people know that the image hasspecial information that only they can read. And we havent even discussed hidden data insideaudio 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 todays technology, information can be smuggled inessentially any type of digital file, including JPEGs or bitmaps, MP3s or WAV files, and MPEGmovies. More than a hundred such steganographic applications are freely available on theInternet. Many of these programs are slick packages whose use requires no significant technicalskills whatsoever. Typically, one mouse click selects the carrier, a second selects the secretinformation to be sent, and a third sends the message and its secret cargo. All the recipient needsis 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 nastyvirus, say), or blueprints. The favored carrier files are the most common ones, like JPEGs orMP3s. This emphasis on popular file formats increases the anonymity of the entire transaction,because these file types are so commonplace that they dont stick out.The one limitation that steganographers have traditionally faced is file size. The rule of thumb isthat you can use 10 percent of a carrier files size to smuggle data. For an ambitioussteganographer, that could be a problem: Imagine an electronic equipment factory employeetrying to explain to the IT department why he has to send his mother a 100-megabyte picture ofthe 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; its enough to conceal every playShakespeare ever wrote.And yet, even with these precautions, conventional steganography still has an Achilles heel: Itleaves a trail. Pictures and other e-mail attachments stored on a companys outgoing e-mail
  10. 10. servers retain the offending document. Anything sent has to bounce through some kind of relayand can therefore be captured, in theory.Steganography poses serious threats to network security mainly by enabling confidentialinformation leakage. The new crop of programs leaves almost no trail. Because they do not hideinformation inside digital files, instead using the protocol itself, detecting their existence isnearly impossible.All the new methods manipulate the Internet Protocol (IP), which is a fundamental part of anycommunication, voice or text based, that takes place on the Internet. The IP specifies howinformation travels through a network. Like postal service address standards, IP is mainly incharge of making sure that sender and destination addresses are valid, that parcels reach theirdestinations, and that those parcels conform to certain guidelines.All traffic, be it e-mail or streaming video, travels via a method called packet switching, whichparcels out digital data into small chunks, or packets, and sends them over a network shared bycountless users. IP also contains the standards for packaging those packets.Lets say youre sending an e-mail. After you hit the Send button, the packets travel easilythrough the network, from router to router, to the recipients in-box. Once these packets reach therecipient, they are reconstituted into the full e-mail.The important thing is that the packets dont need to reach their destination in any particularorder. IP is a "connectionless protocol," which means that one node is free to send packets toanother without setting up a prior connection, or circuit. This is a departure from previousmethods, such as making a phone call in a public switched telephone network, which firstrequires synchronization between the two communicating nodes to set up a dedicated andexclusive circuit. Within reason, it doesnt matter when packets arrive or whether they arrive inorder.As you can imagine, this method works better for order-insensitive data like e-mail and staticWeb pages than it does for voice and video data. Whereas the quality of an e-mail message isimmune to traffic obstructions, a network delay of even 20 milliseconds can very much degradea 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 managesmessages that are less time sensitive. VoIP enables data packets representing a voice call to besplit 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 andreceiving computers. During the second phase, data are sent in both directions in streams ofpackets. Each packet, which covers about 20 milliseconds of conversation, usually contains 20 to160 bytes of voice data. The connection typically conveys between 20 and 50 such packets persecond.Telephone calls must occur in real time, and significant data delays would make for an awkwardconversation. So to ferry a telephone call over the Internet, which was not originally intended forvoice communications, VoIP makes use of two more communications protocols, which had to belayered 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 beenheavily adopted in much of streaming media, such as telephony, video teleconferenceapplications, and Web-based push-to-talk features. To do that, it relies in turn on the UDP.
  11. 11. Because voice traffic is so time critical, UDP does not bother to check whether the data arereliable, intact, or even in order. So in a VoIP call, packets are sometimes stuck in out ofsequence. But thats not a big deal because the occasional misplaced packet wont significantlyaffect the quality of the phone call. The upshot of UDP is that the protocol opens a directconnection 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 previouslysetting up any special transmission channels or data paths. That means its completely private.Compared to old-fashioned telephony, IP is unreliable. That unreliability may result in severalclasses of error, including data corruption and lost data packets. Steganography exploits thoseerrors.Because these secret data packets, or "steganograms," are interspersed among many IP packetsand dont linger anywhere except in the recipients computer, there is no easy way for aninvestigator—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 importanceof both these technologies has resulted in the creation of a new technology for establishingwireless Internet connection through mobile phone, known as Wireless Application Protocol(WAP). However, considering the importance of the issue of data security and especiallyestablishing hidden communications, many methods have been presented. In themeanwhile, steganography is a relatively new method.In this paper, a method for hiddenexchange of data has been presented by using steganography on WML pages (WML standsfor 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 documenttags. 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 aversion of Java language specifically used for small devices, which is called J2ME (Java 2Micro Edition). It was tested on a Nokia series 60 mobile phone.
  12. 12. 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 werent encrypted and anyone with asatellite dish could watch whatever was being shown. Well, this didnt work because all of thenetworks using satellites didnt want the satellite dish owners to be able to receive their satellitefeed 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, encryptit, and send the data over the satellite. If the satellite dish owner had a Videocipher box, and paidfor the channel, then the box would descramble (unencrypted) the signal and return it to itsoriginal, useful form.This was done by using a key that was invertible. It was very important that they key beinvertible, 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.ExampleConsider the message "Red Rum"A message is converted into numeric form according to some scheme. The easiest scheme is tolet space=0, A=1, B=2, ..., Y=25, and Z=26. For example, the message "Red Rum" wouldbecome 18, 5, 4, 0, 18, 21, 13.This data was placed into matrix form. The size of the matrix depends on the size of theencryption key. Lets say that our encryption matrix (encoding matrix) is a 2x2 matrix. Since Ihave seven pieces of data, I would place that into a 4x2 matrix and fill the last spot with a spaceto make the matrix complete. Lets call the original, unencrypted data matrix A.
  13. 13. 18 5 4 0A= 18 21 13 0There is an invertible matrix which is called the encryption matrix or the encoding matrix. Wellcall it matrix B. Since this matrix needs to be invertible, it must be square.This could really be anything; its up to the person encrypting the matrix. Ill use this matrix. 4 -2B= -1 3The unencrypted data is then multiplied by our encoding matrix. The result of this multiplicationis the matrix containing the encrypted data. Well call it matrix X. 67 -21 16 -8X=AB= 51 27 52 -26The 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.ExampleThe 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 candecrypt the message. Other times, the inverse is known by the receiving party. The encryptionmatrix cannot be sent with the data, otherwise anyone could grab the data and decode theinformation. Also, by not having the decoding matrix, someone intercepting the message doesntknow what size of matrix to use.
  14. 14. The receiving end gets the encrypted message and places it into matrix form. 67 -21 16 -8X= 51 27 52 -26 The receiver must calculate the inverse of the encryption matrix. This would be the decryption matrix or the decoding matrix. 0.3 0.2 B-1 = 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 4 0A = X B-1 = 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.
  15. 15. Secrecy in Storage:Secrecy in storage is usually maintained by a one-key system where the user provides the key tothe computer at the beginning of a session, and the system then takes care of encryption anddecryption throughout the course of normal use. As an example, many hardware devices areavailable for personal computers to automatically encrypt all information stored on disk. Whenthe computer is turned on, the user must supply a key to the encryption hardware. Theinformation cannot be read meaningfully without this key, so even if the disk is stolen, theinformation on it will not be useable.Secrecy in storage has its problems. If the user forgets a key, all of the information encryptedwith it becomes permanently unusable. The information is only encrypted while in storage, notwhen in use by the user. This leaves a major hole for the attacker. If the encryption anddecryption are done in software, or if the key is stored somewhere in the system, the system maybe circumvented by an attacker. Backups of encrypted information are often stored in plaintextbecause 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 verylong time. Simple passwords have been used for thousands of years to prove identity. Morecomplex protocols such as sequences of keywords exchanged between sets of parties are oftenshown in the movies or on television. Cryptography is closely linked to the theory and practiceof using passwords, and modern systems often use strong cryptographic transforms inconjunction with physical properties of individuals and shared secrets to provide highly reliableauthentication of identity.Determining good passwords falls into the field known as key selection. In essence, a passwordcan be thought of as a key to a cryptosystem that allows encryption and decryption of everythingthat the password allows access to. In fact, password systems have been implemented in exactlythis way in some commercial products.Credentialing Systems:A credential is typically a document that introduces one party to another by referencing acommonly known trusted party. For example, when credit is applied for, references are usuallyrequested. The credit of the references is checked and they are contacted to determine thecreditworthiness of the applicant. Credit cards are often used to credential an individual to attainfurther credit cards. A drivers 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, manysuch systems are being integrated into the smart-card systems in widespread use in Europe. Asmart-card is simply a credit-card shaped computer that performs cryptographic functions andstores secret information. When used in conjunction with other devices and systems, it allows awide variety of cryptographic applications to be performed with relative ease of use to theconsumer.Electronic Signatures:Electronic signatures, like their physical counterparts, are a means of providing a legally bindingtransaction between two or more parties. To be as useful as a physical signature, electronic
  16. 16. signatures must be at least as hard to forge, at least as easy to use, and accepted in a court of lawas binding upon all parties to the transaction.Electronic Cash:There are patents under force throughout the world today to allow electronic information toreplace cash money for financial transactions between individuals. Such a system involves usingcryptography to keep the assets of nations in electronic form. Clearly the ability to forge such asystem would allow national economies to be destroyed in an instant. The pressure for integrityin such a system is staggering.Graph theoryAlthough a pictorial representation of a graph is very convenient for a visual study, otherrepresentations are better for computer processing. A matrix is a convenient and useful way ofrepresenting a graph to a computer. Matrices lend themselves easily to mechanicalmanipulations. Besides, many known results of matrix algebra can be readily applied to study thestructural properties of graphs from an algebraic point of view. In many applications of graphtheory, such as in electrical network analysis and operations research matrices also turn out to bethe natural way of expressing the problem.
  17. 17. THEOREM 1Two graphs G and G are isomorphic if and only if their incidence matricesA (G) and A (G) differ only by permutations of rows and columns.THEOREM 2If A (G) is an incidence matrix of a connected graphG with n vertices the rank of A (G) is n-1.COROLLARYThe 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 incidencematrix of such a graph will be less than n-1.Therefore the (n-1) by (n-1) reduced incidencematrix of such a graph will not be nonsingular. In other words, the reduced incidence matrix of agraph is nonsingular if and only if the graph is tree.THEOREM 3Let A (G) be an incidence matrix of a connected graph G with n vertices. An (n-1)by (n-1) submatrix of A(G) is nonsingular if and only if the n-1 edges corresponding to the n-1 columns ofthis matrix constitute a spanning tree in G.THEOREM 4Let A and B be, respectively, the circuit matrix and the incidence matrix (of a self – loop – freegraph) whose columns are arranged using the same order of edges. Then every row of B isorthogonal to every row A;That is, A.BT = B.AT = 0CIRCUIT MATRIXLet the number of different circuits in a graph G be q and the number of edges in G be e. Then acircuit 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 asB(G)
  18. 18. The graph in has four different circuits, {a,b},{c,e,g},{d,f,g} and {c,d f,e}.Therefore, its circuitsmatrix is a 4 by (0,1) matrix an shown.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) = , Where B (g1) and B(g2) are the circuit matrices of g1and g2.This observation results from thefact 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.
  19. 19. ConclusionApplications of Matrices are not only Graph theory, Stenography, cryptography. There are alsomany ideas applied in this field secret in banking, communication in military administration,confidential message transduction, computerized lockers, etc are other. Our scholars are stillworking in this field to develop a World Wide secured Communication for all people. Codingand 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.