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Once you know the dominant eigenvector, I recall there was some trick you could do to get the second-dominant eigenvector, by projecting the dominant one out so
by gcr 9y ago
Once you know the dominant eigenvector, I recall there was some trick you could do to get the second-dominant eigenvector, by projecting the dominant one out somehow. How can you repeat power iteration to get all the eigenvectors of a matrix?
- jdonaldson 9y agoYou're talking about Arnoldi Iteration. It's useful in a few cases, but it doesn't scale as well due to the fact the component matrices are no longer sparse.
- thomasahle 9y agoIf you know the first one, you can project everything onto its orthogonal plane.
- qwerty1793 9y agoThe largest eigenvalue of (A - k)^{-1} is 1 / (x - k) where x is the smallest eigenvalue of A that is larger than k. So by using this technique on (A - k)^{-1} for various values of k you can find all of the eigenvalues of A.
- acidburnNSA 9y agoIn practice this is numerically unstable and it's generally better to use methods that use a larger and orthogonal subspace (i.e. Krylov methods) to get multiple eigenvalue/vectors. In nuclear reactor problems you can maybe get about 5 eigenvalues by filtering the larger ones but after that it gets noisy fast. I've gotten up to 1000 good eigenvalues from a large neutron diffusion problem using Arnoldi.