A. Apply the Power Method to estimate the dominant...

a. Apply the Power Method to estimate the dominant eigenvalue and a corresponding eigenvector for the matrix A and initial vector $x_0$ below. Stop at $k=5$. You can use 5 decimal places maximum if you wish (using rounding). $A = egin{bmatrix} 8 & 0 & 12 \ 1 & -2 & 1 \ 0 & 3 & 0 end{bmatrix} ; quad x_0 = egin{bmatrix} 1 \ 0 \ 0 end{bmatrix}$ (You can also choose any other $3 imes 3$ or $4 imes 4$ matrix instead of the matrix and initial vector above, if you wish. Do not use $2 imes 2$ matrix.) b. Use the characteristic equation (as seen in class) to find the exact value of the dominant eigenvalue. Compare with your estimate in part a).

Transcript

00:01 In this question we are given the matrix a which is 8 0 12 1 minus 2 1 0 3 0 and the vector x naught is given which is 1 0 0 and we have to use the power method to find out the dominant eigenvalue and the corresponding eigenvector for this matrix.

00:22 So according to power method we do iterations.

00:26 So let us start with iteration 1.

00:29 In this question we are given k is equal to 5 that means we have to do total 5 iterations and we can go on up to 5 decimal places maximum.

00:40 So in first iteration we will multiply a and x naught.

00:44 So 8 0 12 1 minus 2 1 0 3 0 into x naught is 1 0 0.

00:52 So this will be equal to 8 into 1 plus 0 into 0 plus 12 into 0 which is 8.

00:57 Similarly this will be 1 and this is 0.

00:59 Now out of these three we will take the largest value common.

01:03 So largest is 8.

01:04 So taking 8 common we will left with 1 1 by 8 and 0 1 by 8 means 0 .1 to 5 0.

01:15 That means at the end of first iteration largest eigenvalue eigenvalue is 8 and corresponding corresponding eigenvector eigenvector is equal to 1 0 .1 to 5 0.

01:39 Now similarly we will move on to second iteration now iteration 2.

01:46 In this iteration also we will find out a x naught but this time x naught vector will be the corresponding eigenvector which we received in the previous iteration.

01:57 So 8 0 12 1 minus 2 1 0 3 0 into this vector which is 1 0 .1 to 5 0.

02:06 So on multiplying these two we will get this is 8 this will be 1 minus 2 into 0 .1 to 5 which is 0 .75 and 3 into 1 .0 0 .1 to 5 will be 0 .375.

02:24 Again we will take the largest value common which is again 8.

02:27 So taking 8 common this will be 1 0 .75 divided by 8 is 0 .09375 and this will be equal to 0 .04687.

02:40 So that means at the end of second iteration the largest eigenvalue is 8 and the corresponding eigenvector is this.

02:48 Now we will move on to the third iteration.

02:50 So again we will find out a x naught a is 0 8 0 12 1 minus 2 1 0 3 0 multiplied by the eigenvector which we received in the previous iteration 375 0 .04687...

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