Updating a single eigenvector of a matrix

[inversegamma]

Hello all,

I am facing with the following problem:

I have a large PSD matrix (5000 x 5000), say A_0. I also have all the eigenvectors of this matrix.

I have an iterative rank-one update scheme as follows:

A_t = A_{t-1} + x_tx_t^T.

x_t is a column vector of size 5000.

I am interested in the "first eigenvector" (eigvec with largest eigenvalue) of this matrix at every time instant t.

Is there an option where I don't have to compute the eigenvectors for the whole matrix everytime?

Thank you,

Regards
IG