00:01
All right, so we want to look at the matrix 210 -003 and find its eigenvalues, eigenvectors, and the spectral radius.
00:24
So let's go ahead and find these.
00:28
If you want to find the eigenvalues, we just want to find the characteristic polynomial of a, which is just a determinant of a, minus lambda i and set that equal to 0.
00:43
So we want to first get the characteristic polynomial by taking the determinant of a minus lambda i, which is 2 minus lambda 1 0, 1 2 minus lambda 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.
00:59
If we break it down we can get the following 2 minus lambda times this smaller determinants minus another determined 1 -0 -0 -3 minus lambda.
01:15
And since these are both diagonal matrix determinants, we can just take the product of the diagonals as the determinant.
01:25
So we have 2 minus lambda times 2 minus lambda times 3 minus lambda, minus -lama.
01:34
And so now since we have this expression, we then want to find the roots, but the problem at this point is that this is a pretty complicated expression, so we want to simplify it down some more.
01:45
We notice that both terms have this 3 minus lambda, so if we factor that out, you can get 3 minus lambda times a quantity 2 minus lambda squared minus 1.
01:59
And if we go ahead and expand, we should get the following lambda squared minus 4 lambda plus 4 minus 1.
02:08
This simplifies down again to lambda squared minus 4 lambda plus 3.
02:17
And then we can just factor that to get the following 3 minus lambda multiplied by a lambda minus 3, lambda minus 1.
02:27
And so now we have all of this.
02:32
So we have these three terms, these three factors, and then we set this equal to zero.
02:38
That since this is already in factored form, you can just take.
02:42
The roots lambda is equal to 3 and then another 3 and then 1.
02:48
So we have these eigenvalues lambda 1 is equal to 1 and lambda 2 is equal to 3.
02:56
And so these are our eigenvalues.
02:59
Now to find the eigenvectors we want to do the following.
03:03
So to find that eigenvectors we want to find the null space of these resulting matrices and minus and a minus lambda i so for the first one we have lambda 1 is equal to 1 the the null space of a let me write this smaller a minus i this is equal to the our original matrix is 210, 1200, 003.
03:40
So we basically subtract one from the diagonal.
03:43
So we get, let's go ahead and do, so this says, 1 ,1 ,2, and then this was another one, another one, and then the rest for zeros.
03:58
So then we want to find the resulting null space of this matrix...