maximum eigenvalue of a matrix with rank one update
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There is a (real symmetric) matrix A(t) updated recursively by
A(t)=(t-1)/t*A(t-1)+1/t*a(t)*a(t)',
where a(t) is a column vector.
Suppose the maximum eigenvalue of A(t-1) is known; then is there any efficient method to compute the maximum eigenvalue of A(t)?
Many thanks!
Yang
Réponse acceptée
Roger Stafford
le 21 Juin 2013
Yang, you appear to expect some direct relationship between the maximum eigenvector of A(t-1) and that of A(t), possibly also involving vector a(t). However, the maximum eigenvalue of A(t) is actually dependent on all the eigenvalues and all the eigenvectors of A(t-1), so the relationship would have to be very complicated. I see no better way of determining the maximum eigenvalue of A(t) than calling on the 'eig' or 'eigs' function directly, in spite of its being the result of a recursion.
If vector a were not dependent on t, the limiting case as t approaches infinity, would be just the rank one matrix a*a' itself with its single nonzero eigenvalue and corresponding eigenvector proportional to vector a. However you have presumably used the a(t) notation to indicate that a changes with changing t, so even that is untrue.
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