Okay, so this is a good example of how to diagonalize a matrix. So as we know from last lesson we have a general matrix A, that we want to diagonalize or generally we want to bring this matrix to a power. So we want to say, maybe the matrix A, let's bring it to the tenth power. Now, this might B 10. Very annoying computations and we might not want to be dealing with that. So we can decompose our A matrix into a matrix P times a diagonal matrix, right? Where there are only entries on the diagonal and then P inverse. Okay, so we're left with this decomposition of our original A matrix. And in this example, we have this A, again we might want to bring it to the tenth power, hundredth power, whatever. We want an easier way of doing that. Were given that P is this matrix and D is this diagonal matrix, so we're set up. The only thing we really need now is P inverse. Which from previous lessons we should know, P inverse for a two by two matrix we learned that shortcut, right? So it's, one divided by the determinant. So one divided by the determinant P times, then A and D is swapped -2 swaps with 1. And B and C are negated, so -1 and positive 1. So, 1 and -2 are swapped this 1 turns negative, this negative turns positive. Now the determinant of P is, one times -2. -2 minus 1 times -1, so -2 plus 1 is -1, so 1 divided by -1. This term right here, will just still be -1. So we'll get to the actual inverse matrix is, since this constant -1, we'll just multiply each of these by -1, okay. This is our P inverse matrix. Now, let's see the general strategy that this turns into, when I start raising A to certain powers. So I have, A is = PDP inverse. Let's say, I want A squared, A squared= PDP inverse. That one A, and we multiply by another A, because we're squaring it, PDP inverse, okay. Now, you'll hopefully notice here, that when we multiply, of course we're keeping things in order here. We have P inverse times P. So a matrix times its inverse, will give us the identity matrix. So, it'll cancel out. And so for this, we'll end up with PDD again, times P inverse. And if we keep doing this, we'll notice the trend and you can just do it. If you want to multiply again, you would just multiplied PDP inverse on one of these sides and it'll cancel out just like it did here. And so, for any term A to the k, we'll have PD to the kP inverse. So in this case, that's completely fine. Let's see what it looks like. We'll have A to the k, which again, we don't want to compute for this. We'll just leave it like that, we'll have {1, 1, -1, -2}. That's P, right here, we'll have D to the k. We learned last time that, just through multiplying these, you'll realize that it's true. But any diagonal matrix brought to a power, is just the diagonal entries brought to that same power. So we'll have { 5 to the k, 3 to the k, 0, 0}. And then finally, times P inverse, which we figured out was 21. So no matter what the power is here, it's pretty simple. We've transferred this problem from a somewhat difficult problem of a general matrix to a power, which we kind of don't want to mess with, to an easier problem of this matrix, which this matrix was simple. And now, in the middle we have this diagonal matrix, where it's just a number to a power, which is extremely simple compared to a matrix to a power. Okay, so this is, the general idea behind diagonalizing matrix is, when we see it for this example. Take a difficult problem of many, many major multiplication and turn it into just scale is to a power and then a simple to more matrix multiplications. So that's the idea of it, and next time we'll bring it to a new level a little bit higher up.
