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        riddles >> general problem-solving / chatting / whatever >> Help with quadratic forms
        
(Message started by: knightfischer on Mar 7^th, 2008, 8:22am)

Title: Help with quadratic forms
Post by knightfischer on Mar 7^th, 2008, 8:22am

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.

Title: Re: Help with quadratic forms
Post by Eigenray on Mar 7^th, 2008, 12:34pm

If S is symmetric, then for any two vectors v,w,

<Sv, w> = (Sv)^tw = v^tS^tw = v^tSw = <v, Sw>.

If v, w are eigenvectors, say Sv=http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lambda.gifv, Sw=http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gifw, then

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lambda.gif<v,w> = <http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lambda.gifv, w> = <Sv, w> = <v, Sw> = <v, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gifw> = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gif<v,w>,

so if http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lambda.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ne.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gif, 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.)

Title: Re: Help with quadratic forms
Post by knightfischer on Mar 8^th, 2008, 5:32am

Thanks for your clear explanation.