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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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Nicolas Neuss

Posts: 8
Registered: 6/21/11
Re: How to compute eigenvalues and eigenvectors of real symmetric matrix multiplied by diagonal matrix?
Posted: Apr 26, 2012 9:30 AM
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thobbes71@googlemail.com writes:

> 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



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