 Namos욕. , , .. In Internet matrix product a into x0 which will also be column matrix. Then we take out the largest element as the common factor, this technique is called normalization. It is used to obtain a into x0 is equal to lambda 1 into x1, where lambda 1 is the eigenvalue and x1 is the corresponding eigenvector and this is called the first approximation. Then we compute a into x1 and again take out the largest element as the common factor and put in the form a into x1 is equal to lambda 2 into x2, where lambda 2 is the eigenvalue and the x2 is the corresponding eigenvector and this is called the second approximation. Now this iterative process is continued till two consecutive values of lambda and x are same up to a desired accuracy. The values so obtained are called the largest eigenvalue and the corresponding eigenvector of the given square matrix a. This process of finding the largest eigenvalue and the corresponding eigenvector is called RALGY power method or simply power method. Pause the video and answer the questions that is question is find the product of the matrix the 1 3 first row 4 1 second row minus 2 2 the third row into the matrix minus 1 2 first row 4 1 is the second row. I hope all of you have written the answer. The first matrix is a matrix of order 3 into 2 and the second matrix is of order 2 into 2. Now the product of the two matrix again the matrix which gives the which is of order 3 into 2. Now multiply the first row to the first column that is 1 into minus 1 that is minus 1 plus 3 into 4 that is 12 minus 1 plus 2 that is 11. Now multiply the first row to the second column that is 1 into 2 2 plus 3 into 1 3 that is 3 plus 2 that is 5. Now multiply the second row to the first column 4 into minus 1 plus plus 1 into 4 that is plus 4 minus 4 that cancel which become 0. Now multiply the second row to the second column that is 4 into 2 8 plus 1 into 11 8 plus 1 that is 9. Now multiply the third row to the first column that is minus 2 into minus 1 that is plus 2 plus 2 into 4 that is 8 8 plus 2 that is 10. Now multiply the third row to the second column that is minus 2 into 2 that is minus 4 plus 2 into 1 2 that is minus 2. Now come to an example find the largest eigen value and the corresponding eigen vector of the matrix 1 2 3 4 using the power method by taking the initial eigen vector as the matrix 0 1 transpose. Let a is equal to a square matrix that is 1 2 3 4 and let the initial eigen vector x naught is equal to a column matrix that is 0 1. Now by the power method we have to multiply the a into x naught that is the square matrix and the initial eigen vector which gives the first approximation or iteration that is a into x naught is equal to the matrix 1 2 3 4 into the column matrix 0 1. Now take the product the 1 into 0 0 plus 2 into 1 2 that is a 2 first element of the column matrix. Now multiply the second row to the column matrix that is 3 into 0 0 plus 4 into 1 4 that is in this column matrix that take the largest element as a common factor and is simplifying we get that 4 into the column matrix 0.51 which is equal to lambda 1 into x 1 where the lambda 1 is equal to 4 is a eigen value and x 1 is equal to 0.51 is the corresponding eigen vector. Come to the second iteration a into x 1 which is equal to matrix 1 2 3 4 into x 1 0.51. Now taking the multiplication between the a into x 1 that is 1 into 0.5 plus 2 into 1 that becomes a 2.5 multiply the second row to the column matrix that is 3 into 0.5 plus 4 into 1 that becomes 5.5. Now in this column matrix we take the largest element as a common factor and the simplifying which becomes the 5.5 into matrix 0.45451 which is equal to lambda 2 into x 2 where the lambda 2 is equal to 5.5 is eigen value and x 2 is equal to 0.45451 is a corresponding eigen vector. Now come to the third iteration that is a into x 2 is equal to the matrix 1 2 3 4 into x 2 is 0.45451. Now taking the multiplication between the square matrix a and the column matrix that is first row to the column matrix 1 into 0.4545 plus 2 into 1 that becomes 2.4545. Now multiply the second row to the column matrix that is 3 into 0.4545 plus 4 into 1 that becomes 5.3635. Now in this column matrix now take the largest element as the common factor and simplify which becomes the 5.3635 into the matrix 0.45761 which is equal to lambda 3 into x 3 where the lambda 3 is equal to 5.3635 is the eigen value and x 3 is equal to the matrix the 0.45761 is the corresponding eigen vector. Now come to the fourth iteration that is a into x 3 is equal to the matrix 1 2 3 4 into g into the matrix 0.45761. Now take the multiplication between the square matrix a and the column matrix x 3. Now first multiply the first row to the column matrix that is 1 into 0.4576 plus 2 into 1 that becomes 2.4576. Now multiply the second row to the column matrix that is 3 into 4576 plus 4 into 1 that becomes 5.3728. Now take the largest element as a common factor and simplify which becomes which becomes the 5.3728 into the matrix 0.45741 which is equal to lambda 4 into x 4. Now again taking the fifth iteration that is a into x 4 which is equal to the matrix 1 2 3 4 into matrix 0.45741. Now taking the multiplication here the first row to the column matrix that is 1 into 0.4574 plus 2 into 1 that becomes 2.4574. Now multiply the second row to the column matrix that is 3 into 0.4574 plus 4 into 1 that becomes 5.3722. Now further we take the in the column matrix take the largest element as a common factor that is the 5.3722 into matrix 0.45741 which is equal to lambda 5 into x 5. Now it is observed that the x 4 and x 5 are equal hence we conclude that the largest eigenvalue of a is lambda is equal to 5.3722 and the corresponding eigenvector x is equal to 0.45741 transpose references thank you.