Elementary transformation

Elementary Matrix Transformation- Elementary Operation, Elementary Operation Notation, Elementary Operators, How to perform Elementary Row Operation, How to perform Elementary Column Operation

1. SUNIPA BERA B.Tech cse ( csf ) Roll no:. 2

2. Elementary transformation

3. Acknowledgement Primarily I would thank God for being able to complete this presentation in Success. Then I would like to thank my teacher, whose valuable guidance has been the ones that helped me patch this presentation and make it full proof success her suggestions and her instructions has served as the major contribution towards the completion of the presentation. Then I would like to thank my parents and friends who have helped me with their valuable suggestions and guidance.

4. Elementary Matrix O perations Elementary matrix operation plays an important role in many matrix algebra applications, such as finding the inverse of a matrix and solving simultaneous linear equations. Elementary Operations There are three kinds of elementary matrix operations. Interchange two rows (or columns). Multiply each element in a row ( or columns) by a non-zero number. 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 row as, they are called elementary row operations; and when they are preformed on columns, they are called elementary column operations.

5. 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 i and j R i <_> R j 2. Multiply row i by s, where s≠0 s R i  R j 3. Add s times row i to row j s R i + R j = R j Column Operations 1. Interchange column i and j C i <—> C j 2. Multiply column i by s, where s≠0 s C i  C j 3.Add s times column i to column j s C i + C j  C j

6. Elementary Operation Each type of elementary operation may be performed by matrix multiplication, using square matrices called elementary operators. For example, suppose you want to interchange rows 1 and 2 of matrix A. To accomplish this, you could premultiply A by E to produce B, as shown below. Here, E is an elementary operator. It operates on A to produce the desired interchanged row in B. What we would like to know, of course, is how to find E. Read on.

7. How to Perform E lementary R ow O perators How to Perform Elementary Row Operations To perform an elementary row operation on a  A , an  r  x  c  matrix, take the following steps. To find  E , the  elementary row operator , apply the operation to an  r  x  r   identity matrix . To carry out the elementary row operation, pre multiply   A  by  E . We illustrate this process below for each of the three types of elementary row operations.

8. Interchange two rows. Suppose we want to interchange the second and third rows of A, a 3 x 2 matrix. To create the elementary row operator E, we interchange the second and third rows of the identity matrix I 3 . Then, to interchange the second and third rows of A, pre multiply A by E as shown below

9. Multiply a row by a number. Suppose we want to multiply each element in the second row of Matrix A by 7. Assume A is a 2 x 3 matrix. To create the elementary row operator E, we multiply each element in the second row of the identity matrix I 2  by 7 Then, to multiply each element in the second row of A by 7, we pre multiply A by E

10. Multiply a row and add it to another row. Assume  A is a 2 x 2 matrix. Suppose we want to multiply each element in the first row of A by 3; and we want to add that result to the second row of A. For this operation, creating the elementary row operator is a two-step process. First, we multiply each element in the first row of the identity matrix I 2  by 3. Next, we add the result of that multiplication to the second row of I 2  to produce E. Then, to multiply each element in the first row of A by 3 and add that result to the second row, we pre multiply A by E

11. How to Perform Elementary Column Operation To perform an elementary column operation on A, an  r  x  c  matrix, take the following steps. To find E, the elementary column operator, apply the operation to an  c  x  c   identity matrix . To carry out the elementary column operation, post multiply  A by E. Let's work through an elementary column operation to illustrate the process. For example, suppose we want to interchange the first and second columns of A, a 3 x 2 matrix. To create the elementary column operator E, we interchange the first and second columns of the identity matrix  I 2 . Then, to interchange the first and second columns of A, we postmultiply A by E. as shown below

12. Note that the process for performing an elementary column operation on an  r  x  c  matrix is very similar to the process for performing an elementary row operation. The main differences are: To operate on the  r  x  c  matrix A, the row operator E is created from an  r  x  r  identity matrix; whereas the column operator E is created from an  c  x  c  identity matrix. To perform a row operation, A is  pre multiplied  by E; whereas to perform a column operation, A is  post multiplied  by E.

13. THANK YOU - guided by Naveen Sir