MIT18_06S10_L25 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 25 -- one and -- the lecture on symmetric matrixes. So that's the most important class of matrixes, symmetric matrixes. A equals A transpose. So the first points, the main points of the lecture I'll tell you right away. What's special about the eigenvalues? What's special about the eigenvectors? This is -- the way we now look at a matrix. We want to know about its eigenvalues and eigenvectors and if we have a special type of matrix, that should tell us something about eigenvalues and eigenvectors. Like Markov matrixes, they have an eigenvalue equal one. Now symmetric matrixes, can I just tell you right off what the main facts -- the two main facts are? The eigenvalues of a symmetric matrix, real -- this is a real symmetric matrix, we -- talking mostly about real matrixes. The eigenvalues are also real. So our examples of rotation matrixes, where -- where we got E- eigenvalues that were complex, that won't happen now. For symmetric matrixes, the eigenvalues are real and the eigenvectors are also very special. The eigenvectors are perpendicular, orthogonal, so which do you prefer? I'll say perpendicular. Perp- well, they're both long words. Okay, right. So -- I have a -- you should say "why?" and I'll at least answer why for case one, maybe case two, the checking the Eigen -- that the eigenvectors are perpendicular, I'll leave to, the -- to the book. But let's just realize what -- well, first I have to say, it -- it could happen, like for the identity matrix -- there's a symmetric matrix. Its eigenvalues are certainly all real, they're all one for the identity matrix. What about the eigenvectors? Well, for the identity, every vector is an eigenvector. So how can I say they're perpendicular? What I really mean is the -- they -- this word are should really be written can be chosen perpendicular. That is, if we have -- it's the usual case. If the eigenvalues are all different, then each eigenvalue has one line of eigenvectors and those lines are perpendicular here. But if an eigenvalue's repeated, then there's a whole plane of eigenvectors and all I'm saying is that in that plain, we can choose perpendicular ones. So that's why it's a can be chosen part, is -- this is in the case of a repeated eigenvalue where there's some real, substantial freedom. But the typical case is different eigenvalues, all real, one dimensional eigenvector space, Eigen spaces, and all perpendicular.
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So, just -- let's just see the conclusion. If we accept those as correct, what happens -- and I also mean that there's a full set of them. I -- so that's part of this picture here, that there -- there's a complete set of eigenvectors, perpendicular ones. So, having a complete set of eigenvectors means -- so normal -- so the usual --
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MIT18_06S10_L25 - 18.06 Linear Algebra Spring 2010...

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