28. Similar Matrices and Jordan Form
Okay. This lecture is mostly about the idea of similar matrixes. I'm going to tell you what that word similar means and in what way two matrixes are called similar. But before I do that, I have a little more to say about positive definite matrixes. You can tell this is a subject I think is really important and I told you what positive definite meant -- it means that this -- this expression, this quadratic form, x transpose I x is always positive. But the direct way to test it was with eigenvalues or pivots or determinants. So I -- we know what it means, we know how to test it, but I didn't really say where positive definite matrixes come from. And so one thing I want to say is that they come from least squares in -- and all sorts of physical problems start with a rectangular matrix -- well, you remember in least squares the crucial combination was A transpose A. So I want to show that that's a positive definite matrix. Can -- so I -- I'm going to speak a l...