00:01
In this question, we are going to orthogonally diagonalize the matrix a.
00:07
So the first thing to do is to find eigenvalues.
00:15
So we need to solve for the character reciprocal to be 0.
00:21
So 6 minus lambda, negative 2, 2, 3 minus lambda, negative 1, 2, 2, negative 1, 3 minus lambda.
00:33
If we expand this determinant, what we will get is negative negative cube plus 12 lambda square minus 36 lambda plus 32.
01:02
This can be factored as negative lambda minus 2 square, lambda minus 8.
01:16
So the eigenvalues of a are 2, 2, 8.
01:27
So next we compute eigenvectors.
01:37
So we first look at eigenvalue 2, so we compute a minus 2i, which is 4, negative 2, 1, negative 1, 2, 1.
01:53
1.
01:56
We are going to compute its kernel so we do role operations.
02:01
The first row, it can be divided by 2.
02:14
And we know that now first and third row are the same and the second row is just have a negative sign.
02:21
So we can eliminate the second and third row completely.
02:29
So we'll get 2, negative 1, 1 and 0000000, 0000.
02:38
So now we know that the kernel of a minus 2i, this is spanned by two vectors.
02:49
And we know that this two is pivot.
02:54
And so the second and third variables are non -pivot variables.
03:01
So we consider it to be 1 -0 and 0 for the non -pivot variables...