  We define a new diffusive matrix model converging towards the $\beta$-Dyson
Brownian motion for all $\beta\in [0,2]$ that provides an explicit construction
of $\beta$-ensembles of random matrices that is invariant under the
orthogonal/unitary group. We also describe the eigenvector dynamics of the
limiting matrix process; we show that when $\beta< 1$ and that two eigenvalues
collide, the eigenvectors of these two colliding eigenvalues fluctuate very
fast and take the uniform measure on the orthocomplement of the eigenvectors of
the remaining eigenvalues.
