Interlacing eigenvectors of large Gaussian matrices

We consider the eigenvectors of the principal minor of dimension n < N of the Dyson Brownian motion in R-N and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large n,N limit with n/N tending to a fixed ratio q, for any initial symmetric matrix A. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the Gaussian orthogonal ensemble case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where A has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion.