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Topic: Help with quadratic forms (Read 1712 times) |
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knightfischer
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Help with quadratic forms
« on: Mar 7th, 2008, 8:22am » |
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I'm trying to understand how to derive the orthonormal matrix Q for a symmetric matrix S, such that S = QDQt, where Qt is the transpose=inverse of the orthonormal matrix Q.
I know how to obtain the eigenvalues, and how to get a set of eigenvectors, but these eigenvectors are not necessarily orthogonal. I know how to make an orthogonal vector orthonormal.
So, given a set of eigenvectors for a symmetric matrix, how do I obtain the orthogonal matrix Q?
Can anyone help me with this. I know it is a basic idea in Linear Algebra, but I cannot find a clear explanation in a text or on the web.
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Eigenray
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Re: Help with quadratic forms
« Reply #1 on: Mar 7th, 2008, 12:34pm » |
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If S is symmetric, then for any two vectors v,w,
<Sv, w> = (Sv)tw = vtStw = vtSw = <v, Sw>.
If v, w are eigenvectors, say Sv= v, Sw= w, then
<v,w> = <v, w> = <Sv, w> = <v, Sw> = <v, w> = <v,w>,
so if  , then we must have <v,w>=0. That is, two eigenvectors from distinct eigenspaces must be orthogonal. So it suffices to find an orthonormal basis of each eigenspace (for example, by Gram-Schmidt), because the union of these bases will still be orthonormal.
(More generally, if S is Hermitian, all the eigenvalues with be real, and eigenvectors from distinct eigenspaces will again be orthogonal.)
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| « Last Edit: Mar 7th, 2008, 12:37pm by Eigenray » |
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knightfischer
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Re: Help with quadratic forms
« Reply #2 on: Mar 8th, 2008, 5:32am » |
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Thanks for your clear explanation.
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