maximum eigenvalue of a matrix with rank one update

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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Yang 2013-6-25

Roger, thank you very much for your response.

I have also realized that there is not a closed-form expression connecting the largest eigenvalue of A(t) and A(t-1), unless some special structures are assumed.

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