This concept of shooting rays through a paraxial system with the refraction and transfer equations is so important. We're going to do it again with a slightly different mathematics. And that might seem redundant. But by recognizing that this is a linear system, and therefore, we can cascade up these calculations with matrices, we're going to go up a level in the sophistication and power of what we can do. Now, I have a warning for you here. The sort of level of mathematical sophistication and the density of the calculations we're going to do is about to go up a lot. So walk through this carefully, and make sure that you're following. Because we're going to very rapidly be able to start drawing some relatively sophisticated conclusions from these matrices that we're about to define. So let's take, for example, our refraction equation. And say, hey, maybe what we're doing is we're simply pushing a two element vector that describes the right. And again, we're in a plane here. It would be a four-element vector if we have both planes to worry about. Now let's make life simple and just deal, stay in a single plane. So there's two elements, two numbers we care about, y and u. So why don't we define a little two-element vector y, u, that's the ray height and paraxial ray angle, going into a surface k, because there's no prime, only u. And then the refraction equation would tell us how we get to the other side of the lens, where of course, we now have the same ray height but a new ray angle, u prime. And we can simply rearrange that equation. You can see, for example, across the top here, y will equal y, as it should. But the bottom line is our refraction equation. So we'll call that little 2x2 matrix, which is typically called an ABCD matrix. It's not very inventive, I'm sorry, A, B, C, D, the four elements. We'll call this particular one R, for refraction. And it's the kth refraction equation that describes how to refract across lens k. And of course, similarly, we can look at the transfer equation. And we can take now a ray height and a ray angle out of a surface and get to the next surface where we have a new ray height and a new ray angle. Of course, this ray angle going into the k + 1 surface had better be equal to the ray angle that came out of the k surface. And again, that's what the 0 one down here mean. So this is just the transfer equation wrapped up in a 2x2 matrix, we'll call the transfer matrix. And why this is interesting of course, is now rather than having to use a tabular method or one at a time, we write down each of the equations, now we can just cascade our matrices. And immediately describe entire systems of transfer and refraction by this cascade of these matrices. Now, let me use this example right here for sort of a typical system. If we started out at some surface 1, and let's say that's, yeah, this surface right here, then we'll have to refract across the lens, number 1, so we'll write R1. Then we'll transfer past lens 1, and we know we're transferring past lens 1, because here we're getting from yk to yk + 1, so this is now 2. So just keep track of what these matrices refer to where you are. So we would then write down, sorry I'm up here. So we'd refract to surface 1, then we transfer past surface 1, then we'd refract to surface 2. Note that I'm going forward, it goes left to right. But I have to write the matrices backwards. And that's because I start with my initial vector. Then I refract, then I transfer, then I refract. And so the matrices are written in the reverse order that light is going and that's just a common confusion. So let's define as a surface standard notation, that the object is at surface 0. There are a capital K number of lenses. So one, two, three, four, five, six, little k being some arbitrary index, and finally, a capital K being the last lens element in the system. And that means that the image is a capital K + 1, and let's just adopt that as a standard system. If we have an imaging system then, we can immediately define two incredibly useful matrices. And we're going to use these to abstract a lot of important properties out of a system. This system matrix defines and describes everything that's inside the optical system, except the object and the image distances. So we start with ray height 1 and ray angle u1 going into surface 1. We can tell it's going into because there's no prime here. We refract across lens 1, go through a total of K surfaces, and we just come out of the last element K. So we have the ray height yK and the ray angle prime that is coming out of surface K. And so all of those cascaded together, we'll call M in the system matrix. And the important point about that, and this is pretty magical now, is now we can take any ray, y1, u1, n, and multiply it by this 2x2 matrix. And we can predict the way ray height and angle coming out. It's a linear system and so a 2x2 matrix can fully capture its behavior. And then finally, there's another matrix which will be useful which is called the conjugate matrix. And that's where we simply add on the object and image distances now traveling from, let's do here, traveling from the object height and the ray angle out of the object, u0 prime. And going all the way to the image, the height yK+1, right there, and the angle uK+1. So it's exactly the same as the system matrix. But we wrap the two transfers on the beginning and end that get us to the system and out of the system to the object. So these two matrices carry a lot of information about the system and we'll use those now to start writing properties of the system as defined by these two matrices.