  Gaussian states of quantum oscillators are fully characterized by the mean
values and the covariance matrix of their quadrature observables. We consider
the dynamics of a system of oscillators subject to interactions, damping, and
continuous probing which maintain their Gaussian state property. Such dynamics
is found in many physical systems that can therefore be efficiently described
by the ensuing effective representation of the density matrix $\rho(t)$. Our
probabilistic knowledge about the outcome of measurements on a quantum system
at time $t$ is not only governed by $\rho(t)$ conditioned on the evolution and
measurement outcomes obtained until time $t$, but is also modified by any
information acquired after $t$. It was shown in [Phys. Rev. Lett. 111, 160401
(2013)] that this information is represented by a supplementary matrix, $E(t)$.
We show here that the restriction of the dynamics of $\rho(t)$ to Gaussian
states implies that the matrix $E(t)$ is also fully characterized by a vector
of mean values and a covariance matrix. We derive the dynamical equations for
these quantities and we illustrate their use in the retrodiction of
measurements on Gaussian systems.
