# The eigenvectors of Gaussian matrices with an external source

We consider a diffusive matrix process (*X _{t}*)

_{t≥0}defined as

*X*:=

_{t }*A*+

*H*where

_{t}*A*is a given deterministic Hermitian matrix and (

*H*)

_{t}_{t≥0}is a Hermitian Brownian motion. The matrix

*A*is the "external source" that one would like to estimate from the noisy observation

*X*at some time

_{t}*t*> 0. We investigate the relationship between the non-perturbed eigenvectors of the matrix A and the perturbed eigenstates at some time t for the three relevant scaling relations between the time t and the dimension

*N*of the matrix

*X*. We determine the asymptotic (mean-squared) projections of any given non-perturbed eigenvector |

_{t}*ψ*, associated to an eigenvalue

^{0}_{j}⟩*aj*of

*A*which may lie inside the bulk of the spectrum or be isolated (spike) from the other eigenvalues, on the orthonormal basis of the perturbed eigenvectors |

*ψ*. We derive a Burgers type evolution equation for the local resolvent (

^{t}_{i}⟩, i ≠ j*z*−

*X*)

_{t}^{−1}

*, describing the evolution of the local density of a given initial state |*

_{ii}*ψ*. We are able to solve this equation explicitly in the large

^{0}_{j}⟩*N*limit, for any initial matrix

*A*. In the case of one isolated eigenvector |

*ψ*, we prove a central limit Theorem for the overlap

^{0}_{j}⟩*⟨ψ*. When properly centered and rescaled by a factor

^{0}_{j}|ψ^{t}_{j}⟩*√N*, this overlap converges in law towards a centered Gaussian distribution with an explicit variance depending on

*t*. Our method is based on analyzing the eigenvector flow under the Dyson Brownian motion.