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GILBERT STRANG: Moving now to the second half of linear algebra. It's about eigenvalues and eigenvectors. The first half, I just had a matrix. I solved equations. The second half, you'll see the point of eigenvalues and eigenvectors as a new way to look deeper into the matrix to see what's important there. OK, so what are they? This is a big equation, S time x. So S is our matrix. And I've called it S because I'm taking it to be a symmetric matrix. What's on one side of the diagonal is also on the other side of the diagonal. So those have the beautiful properties. Those are the kings of linear algebra. Now, about eigenvectors x and eigenvalues lambda. So what does that equation, Sx equal lambda x, tell me? That says that I have a special vector x. When I multiply it by S, my matrix, I stay in the same direction as the original x. It might get multiplied by 2. Lambda could be 2. It might get multiplied by 0. Lambda there could even be 0. It might ge...

JUDY HOYT: So we're moving along pretty quickly with the first part of the course here. I'm hoping you'll be reading Chapter 3 of the textbook at this point, which is on-- covers crystal growth, wafer fabrication, and some of the basic properties of silicon. I'm going to finish up Chapter 3-- the lectures on Chapter 3 today. And then I'm going to start Chapter 4, so you can start that as well. Chapter 4 is relatively easy reading, not tremendously mathematical. It's about how we-- although it's a very important topic. It's about how we clean wafers and how we get our impurities in wafers. So the last lecture, let me just remind you what we talked about. We talked about cubic crystal structures. We discussed wafer manufacturing, basic characteristics of Czochralski wafers. And we'll continue on with that a little bit today. We presented a very simple mathematical model differential equation for-- based on heat flux for modeling the Czoch...