  We consider a rank-one symmetric matrix corrupted by additive noise. The
rank-one matrix is formed by an $n$-component unknown vector on the sphere of
radius $\sqrt{n}$, and we consider the problem of estimating this vector from
the corrupted matrix in the high dimensional limit of $n$ large, by gradient
descent for a quadratic cost function on the sphere. Explicit formulas for the
whole time evolution of the overlap between the estimator and unknown vector,
as well as the cost, are rigorously derived. In the long time limit we recover
the well known spectral phase transition, as a function of the signal-to-noise
ratio. The explicit formulas also allow to point out interesting transient
features of the time evolution. Our analysis technique is based on recent
progress in random matrix theory and uses local versions of the semi-circle
law.
