Linear Algebra – Eigenbasis Example

Now that we've walked through the theory of age bases and diagonalization let's have a go in a simple 2d example, where we can see the answer graphically, so we'll know whether our method has worked as expected. Consider the transformation matrix, T equals 1 1, 0 2, hopefully you'd feel fairly comfortable at this point in drawing the transformed square and vectors that were used in the previous examples. As the first column is just 1 0. This means that our I hat vector will be unchanged. However, The second column tells us that J hat.

The second vector will be moving to the point 1 2, let's also consider the orange diagonal vector to point 1 1 multiplying through gives us 1 1, 0 2 multiplied by 1 1. So thinking about rows times Cole's we're going to get 1 plus 1 and 0 plus 2, which gives us 2 2, it's interesting to consider that this particular transform could be decomposed into a vertical scaling by a factor of 2 and then a horizontal shear by a half step. Because we've chosen such a simple. Transformation, hopefully you've already spotted the eigenvectors and can state their eigenvalues. These are at lambda equals 1.

Our eye, fen vector is 1 0 and lambda equals 2, our vector equals 1 1. Now, let's. Consider what happens to the vector? Minus 1 1 when we apply T? So 1 1, 0 2 applied to minus 1 1. This time is going to equal rows times Cole's, minus 1, plus 1 and 0 plus 2, which equals 0 2.

And if we apply teeth again, we're going to get the following 1 1, 0 2 applied to 0 2, which is going to equal. Rows times cause again, 0 + 2 + 0 + 4. So this thing finally is going to equal to 4 now instead, if we were to start by finding T's squared, so T squared is going to equal this Vic.

This matrix multiplied by itself. So applying rows x, coals, we're going to get 1 times 1 times 1 times 0 so that's, one rows times cause here, we're going to get three rows times cos here, we're going to get a 0 those times' cos here, we're going to get a 4. Now we can apply this to our vector and see if we get the same result. So.1 3 0, 4 x. - 1 1 is going to equal rows times Cowley, so we're going to get - 1 + 3 + 0 + 4, which of course equals to 4.

We can now try this whole process again. But using our eigenbasis approach, we've already built our conversion matrix C from our age vectors. So C is going to equal 1 0 1 1, but we're now going to have to find its inverse. However, because we've picked such a simple, 2x2 example, it's possible just to write this inverse down directly by considering that C would just be a horizontal. Shear one step to the right so C inverse must just be this. Same shift back to the left again.

So C inverse is going to equal 1 - 1 0 1 it's worth noting that despite how easy this was I would still always feed this to the computer instead of risking making still mistakes. We can now construct our problem. So T squared is going to equal see, d squared C inverse, which, of course, in our case is going to equal 1 1, 0 1 multiplied by our diagonal matrix, which is going to be 1 and 2 and that's all squared. Multiplied by C inverse, which is 1 minus 1 0 1 working this through we can see that let's keep this first matrix 1 1, 0 1 and work out this bit. So we'll say, ok, this is going to be 1 and 4 on the diagonal 1. Minus 1, 0 1 and let's work out these two matrices here.

So we've got 1 1, 0 1 x. So we're doing rows times calls in each case. So for example, 1 0 times 1 0, you get a 1 here go to the second row in the first column. We get a 0 there first row and second column we're going to get a minus 1 there and. The second row and the second column we're going to get 4 here.

Okay. And then working it through one more step we're going to see that we get more grinding. We get first row. First column 1 second row, first column, 0 first row. Second column is 3 and second row. Second column is 4 and applying this to the vector, minus 1 1 we're going to get something like this.

So 1 3 0 4 applied to minus 1 1. It's going to be rows times Cole's. So minus 1 plus 3 and 0 plus 4, which equals to 4, which pleasingly enough is the.

Same result as we found before now, there is a sense in which for much of mathematics once you're sure that you've really understood a concept, then because of computers you may never have to do this again by. However, it is still good to work through a couple of examples on your own just to be absolutely sure that you get it. There are of course, many aspects of age theory that we haven't covered in this short video series, including diagnosable, matrices and complex age vectors. However, if you. Are comfortable with the core topics that we've discussed then you're already in great shape in the next video we're going to be looking at a real-world application of age theory to finish off this linear. Algebra course, this is a particularly famous application, which requires abstraction away from the sort of geometric interpretations that we've been using so far, which means that you'll be taking the plunge and just trusting the math see you then.

Dated : 10-Apr-2022