MIT18_06S10_L32 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 32 OK, here we go with, quiz review for the third quiz that's coming on Friday. So, one key point is that the quiz covers through chapter six. Chapter seven on linear transformations will appear on the final exam, but not on the quiz. So I won't review linear transformations today, but they'll come into the full course review on the very last lecture. So today, I'm reviewing chapter six, and I'm going to take some old exams, and I'm always ready to answer questions. And I thought, kind of help our memories if I write down the main topics in chapter six. So, already, on the previous quiz, we knew how to find eigenvalues and eigenvectors. Well, we knew how to find them by that determinant of A minus lambda I equals zero. But, of course, there could be shortcuts. There could be, like, useful information about the eigenvalues that we can speed things up with. OK. Then, the new stuff starts out with a differential equation, so I'll do a problem. I'll do a differential equation problem first. What's special about symmetric matrices? Can we just say that in words? I'd better write it down, though. What's special about symmetric matrices? Their eigenvalues are real. The eigenvalues of a symmetric matrix always come out real, and there always are enough eigenvectors. Even if there are repeated eigenvalues, there are enough eigenvectors, and we can choose those eigenvectors to be orthogonal. So if A equals A transposed, the big fact will be that we can diagonalize it, and those eigenvector matrix, with the eigenvectors in the column, can be an orthogonal matrix. So we get a Q lambda Q transpose. That, in three symbols, expresses a wonderful fact, a fundamental fact for symmetric matrices. OK. Then, we went beyond that fact to ask about positive definite matrices, when the eigenvalues were positive. I'll do an example of that. Now we've left symmetry. Similar matrices are any square matrices, but two matrices are similar if they're related that way. And what's the key point about similar matrices? Somehow, those matrices are representing the same thing in different basis, in chapter seven language. In chapter six language, what's up with these similar matrices? What's the key fact, the key positive fact about similar matrices? They have the same eigenvalues. Same eigenvalues.
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So if one of them grows, the other one grows. If one of them decays to zero, the other one decays to zero. Powers of A will look like powers of B, because powers of A and powers of B only differ by an M inverse and an M way on the outside. So if these are similar, then B to the k-th power is M inverse A to the k-th power M. And that's why I say, eh, this M, it does change the eigenvectors, but it doesn't change the eigenvalues. So same lambdas. And then, finally, I've got to review the point about the SVD, the Singular Value Decomposition. OK. So that's what this quiz has got to cover, and now I'll just take problems from earlier
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MIT18_06S10_L32 - 18.06 Linear Algebra Spring 2010...

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