Application of Matrices in real life | Matrices application | The Matrices
- 1. Matrices Its applications in real life..
- 2. Matrices can dominate vectors by giving them new values Matrices and Vectors01 Eigen vector related problem in daily life. Eigen Vector02 Coding and Encoding Cryptography03
- 3. Vector and Matrix • Matrices (for example Square matrix of order 2) can rotate any vector in 2-D plane resulting in new value and direction. • If we write vectors in matrix form and multiply them with any matrix the resultant would be matrix in same plane but may be of different magnitude and direction. • For example:
- 4. • In the above example you can see that we have a vector (1,1) in X-Y plane and when we write it in matrix form and multiply it by a matrix resultant is a vector of different magnitude and direction. • Again we can design a matrix in many way to manipulate a given vector as mentioned below. When any vector is multiplied by the given matrices its magnitude became double but direction remain same. When any vector is multiplied by given matrices its rotate 90 degree in anticlockwise direction.
- 5. Now question arises where is the application ? We have discussed what matrices do to vector but here we are for applications of it and so far nothing related to it discussed. As a major application you may tried to solve linear equation problems in matrix as below but what if I say that all you were doing was just to get a vector. Wait I will show you. Consider the example taken from previous slide- Just a basic Matrix linear problem all we have to do is find x and y. We will assume a matrix of 2x1 as of x and y and multiply with a matrices and solve for it but ever you think upon it in another way..... All we are doing is a transformation of vector in backward direction moreover we are trying to find inverse of A. If we assume as unknown matrix to be A then this can be explained as below
- 6. Simply we are doing following transition. The red marked is the RHS side that is (1,3) vector and blue one is what we are trying to find out. It's simply an Inverse matrix. Since A inverse of (1,3) is (1,1) hence is the result
- 7. Eigen Vector Any vector that is scaled by matrix is called eigen vector and factor by which it is scaled is called eigen value. In simple language it is not rotated in plane and only its magnitude changes. This simple definition can be used in many ways. Let's understand from a example. Consider the situation of Zombie attack on a place. The place has been isolated from surrounding and hence it can not spread to others. Since there is no cure for it only medicines are given and it converts nearly 10 percent of them back to zombie. We need to find when a balanced situation would arise or what happens in long run. Lets consider an ideal situation of equal percentage of zombies and humans. Since after end of each hour their value changes and hence we cannot solve it again and again taking values. If we go through matrices..........
- 8. Lets consider a situation after an hour: Number of Humans- • Since every hour 20 percent of humans convert into zombies hence 80 percent remains as humans. • Also 10 percent of Zombies are converted back to Humans. Therefore Humans after one hour are 80 percent of Humans and 10 percent of Zombies, Mathematically 0.8(150) + 0.1(150) = 135 Number of Zombies- • Since 10 percent of Zombies are converted into Humans hence 90 percent remains as it is. • Also every hour 20 percent of Humans are converted into Zombies. Therefore zombies after one hour are 0.9(150) + 0.1(150) = 165 Again after an hour we will perform same calculations and get the results as:
- 9. Now if we go through Matrix method: If we represent those percentages as matrices then we get a linear equation as given below: Where, v and are initial values of Humans and Zombies population and and are population after an hour. If we plot a graph between their different values keeping Humans on X-axis and Zombies on Y-axis :
- 10. After examining above graph we can conclude that all vectors are aligning themselves in a particular fixed direction and that is the Eigen vector direction. When you compute Eigen Value for above then resultant will be one and therefore we can compute Eigen Vector which is (1,2). Therefore answer will be 100 for Humans and and 200 for Zombies.
- 11. Cryptography • Cryptography mainly consists of Encrypton and Decryption. • Cryptography, is concerned with keeping communications private. • 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. • They both required some secret information called as key. How it works? • One type of code, which is extremely difficult to break, makes use of a large matrix to encode a message. • The receiver of the message decodes it using the inverse of the matrix. • This first matrix, used by the sender is called the encoding matrix and its inverse is called the decoding matrix, which is used by the receiver.
- 12. ATTACK ON CHINA For example if Indian government wants to send this message to all its units in case of extreme border disputes than there is need of some way to keep secret all this message and hence they can send message in unreadable form using Matrices. We assign a number for each letter of the alphabet. Such that A is 1, B is 2, and so on. Also, we assign the number 27 to space between two words. Thus the message becomes: A T T A C K . O N . C H I N A 1 20 20 1 3 12 27 15 14 27 3 8 9 14 1 For sending first of all encoding of message take place . Let the encoding matrix be: -3 -3 -4 0 1 1 4 3 4
- 13. ENCODING Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1 vectors: 1 1 27 27 9 20 3 15 3 14 20 12 14 8 1 • Note that it was necessary to add a space at the end of the message to complete the last vector. • We encode the message by multiplying each of the above vectors by the encoding matrix. • We represent above vectors as columns of a matrix and perform its matrix multiplication with the encoding matrix -3 -3 -4 0 1 1 4 3 4 20 3 15 3 14 1 1 27 27 9 20 12 14 8 1
- 14. We Get -143 -60 –182 -122 –73 21 13 41 35 10 163 63 197 125 87 • The columns of this matrix give the encoded message. • Encoding is complete.
- 15. TRANSMISSION The message is transmitted in a linear form: -143, -60, –182, -122, –73, 21, 13, 41, 35 ,10, 163, 63, 197, 125, 87
- 16. DECODING To decode the message: • The receiver writes this string as a sequence of 3 by 1 column matrices and repeats the technique using the inverse of the encoding matrix. • The inverse of this encoding matrix is the decoding matrix. − 4 − 3 − 3 4 4 3 1 0 1 • To decode the message, perform the matrix multiplication: − 4 − 3 − 3 4 4 3 1 0 1 -143 -60 –182 -122 –73 21 13 41 35 10 163 63 197 125 87
- 17. Matrix obtained is: 20 3 15 3 14 1 1 27 27 9 20 12 14 8 1 Message is decoded as: 1 20 20 1 3 12 27 15 14 27 3 8 9 14 1 A T T A C K . O N . C H I N A Message received as : ATTACK ON CHINA
- 18. Bibliography • www.slideshare.com • https://youtu.be/rowWM-MijXU • www.linkldn.com • https://matrix.reshish.com/
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