All right, one more eigenvalue computation example.
This one's a little more interesting because the matrix is actually quite a bit larger.
But it has a special structure that's going to make it actually kind of easy to figure
out what the determinant is.
So here's our matrix A, it consists of seven rows and seven columns.
So it is a seven by seven matrix.
A lot of entries, but notice one very important thing-- many of the entries are 0.
And in fact, all of the entries below the diagonal here are in fact 0.
So this is what we call an upper triangular matrix because all of the non-zero values
are on the diagonal or above the diagonal.
So this is the matrix A, and we are going to find the eigenvalues of A. So in general,
when asked to find the eigenvalues we think, ah, I need to find the characteristic polynomial
of the matrix.
And then I need to set that characteristic polynomial to 0 and find the roots of when
that polynomial is equal to 0.
And that by definition is the collection of eigenvalues that we're looking for.
So if we do that for this particular example, that means we need to find when the determinant
of A minus lambda I is equal to 0, just like we did in the previous examples.
So let's go ahead and write out what that looks like.
So I need to find the determinant.
So these vertical lines are notation for determinant.
So if you put vertical lines around a matrix that means the determinant of the matrix inside.
So I'm going to compute the determinant of the matrix A minus lambda I.
Remember, lambda I is an identity matrix of the appropriate size that's been scaled by
lambda.
Since we're working with a seven by seven matrix, I need to make I a seven by seven
matrix, multiply it by lambda.
So it's going to have lambda down the diagonals.
So I'm going to have 2 minus lambda here in my first entry and nothing happens right here.
Then here I'm going to have minus 1 minus lambda because I'm subtracting lambda off
of the diagonal element.
And then the other non-diagonal terms are unchanged, and so on.
All the way down.
Essentially every single entry along the diagonal is going to be the original entry minus lambda,
just like we've seen in the previous examples.
And this is now the matrix that I need to compute the determinant of.
So one thing we noted earlier, is that this is a triangular matrix.
In fact, it's an upper triangular matrix because all of the non-zero values are here above
the diagonal.
There's a very nice property of triangular matrices.
If you were to do co-factor expansion about this first row you would see that many of
the entries are going to end up being 0.
As we do our expansion all these zeros down here are going to end up or result in a lot
of zero terms.
And if you choose your co-factor expansion wisely, you'll see what happens is that the
determinant ends up being just the product of all of these diagonal elements.
So the rule-- we haven't proven here, but if you understand the reasoning-- for diagonal
matrices is that you just have to multiply these diagonal elements.
So we can just write down the determinant right away.
The determinant is just 2 minus lambda times a negative 1 minus lambda times 1/2 minus
lambda.
We just have to multiply all of these diagonal elements times each other.
So this right here is the characteristic polynomial.
If we set that equal to 0, we can now find the eigenvalues.
Now we can do this just by inspection.
One of the eigenvalues is 2, one of the eigenvalues is a negative 1, one the eigenvalues is a
1/2, and so on.
So this is just a write down at this point.
lambda 3 is a 1/2, lambda 4 is 9, lambda 5 is 3, lambda 6 is a negative 6.
So these are the eigenvalues of this matrix.
So a much larger matrix in this case, but it had a very special structure.
So we actually didn't have to do that much algebra.
We were just able to use this very important fact anytime you're working with a triangular
matrix-- so that's a matrix that has all of its non-zero values on or above the diagonal,
or all of its non-zero values on or below the diagonal.
So it's very possible to write down something similar, that's what we would call a lower
triangular matrix.
If you have a matrix like that, the determinant is trivial to compute, which means identifying
the eigenvalues is also trivial as well.