
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: > thobbe...@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
Thanks a lot, Nicolas. This was most helpful.
Tom

