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Topic: How to compute eigenvalues and eigenvectors of real symmetric matrix
multiplied by diagonal matrix?

Replies: 7   Last Post: Apr 27, 2012 4:57 PM

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 thobbes71@googlemail.com Posts: 4 Registered: 4/25/12
Re: How to compute eigenvalues and eigenvectors of real symmetric
matrix multiplied by diagonal matrix?

Posted: Apr 27, 2012 2:15 AM

On 26 Apr., 15:30, Nicolas Neuss <lastn...@scipolis.de> wrote:
> > Hello,
>
> > there exist efficient algorithms to compute the eigenvalues and
> > eigenvectors of a real symmetric matrix A. But how about a real
> > symmetric matrix which has been multiplied by a diagonal matrix D (all
> > diagonal elements are real and >0), thus destroying the symmetry of A?

>
> > Are eigenvectors and eigenvalues of A and of D*A related in a way
> > which can be exploited to efficiently compute the eigensystem of D*A?

>
> Almost.  Let D^{1/2} denote the square root of D.  Then D*A is similar
> to B=D^{1/2}*A*D^{1/2} (which is symmetric) and has therefore the same
> eigenvalues.  If you now have an eigenvector x of B it gives you an
> eigenvector y=D^{1/2}*x of D*A with eigenvalue lambda because of
>
> D*A*y = D^{1/2} * D^{1/2} * A * D^{1/2} * x
>       = D^{1/2} * B * x
>       = lambda * y
>
> Nicolas

Thanks a lot, Nicolas. This was most helpful.

Tom