1. For a transformation matrix A = [7 4; 4 -1] (1) Find the...

1. For a transformation matrix A = [7 4; 4 -1] (1) Find the eigenvalues and eigenvectors of matrix A. (2) Determine the two normalized eigenvectors by using (?x(i))T?x(i) = 1. (3) For this symmetric matrix A, use orthogonal property for eigenvectors to check your answer. (36 points, credits you will get come from your problem-solving procedure, results directly came from the calculators do not count.)

Transcript

00:01 So let us start with the concept which we are going to use here for this question.

00:06 So if you have been given a matrix you have been asked to find the eigen values and eigenvectors you have to use the formula a minus lambda i its determinant to equate it to zero.

00:19 So we are given the matrix a is equals to 7 4 4 minus of 1 with the elements and obviously the identity matrix will be 1 0 0 1.

00:30 So let us find for the first we have to find the eigenvalues and eigenvector for this given matrix.

00:36 So let us apply a minus lambda of i so this can be equals to 7 4 4 minus of 1 negative of lambda of identity matrix 1 0 0 1.

00:48 So this will comes out equals to the 7 minus lambda 4 4 minus 1 minus lambda.

00:59 Now let us find this determinant of this 7 minus lambda 4 4 minus 1 minus lambda and this we have to equate it to 0.

01:08 Expand it along the first row will become 7 minus lambda times of minus 1 minus lambda negative of 4 into 4 will become 16 that will be goes to 0.

01:18 We can form this like an equation that is lambda square plus lambda minus of 7 lambda minus 7 minus 16 is equals to 0.

01:29 This implies that the lambda square minus 6 lambda minus 23 equals to 0.

01:35 This will gives us the lambda is equals to minus 6 plus minus under root over b square minus 4 c formula i am using that is 6 square minus minus will get 4 into 23 divided by 2 times of a is nothing but 1.

01:51 So you will get the required values of lambda is equals to 17 .31 and lambda 2.

01:57 Okay this is lambda 1 this is lambda 2 minus 5 .3.

02:01 So these are what the eigenvalues which we were asked to find out.

02:06 Now also we have to find the eigenvectors...

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