  The characteristics of the model dynamics are critical in the performance of
(ensemble) Kalman filters. In particular, as emphasized in the seminal work of
Anna Trevisan and co-authors, the error covariance matrix is asymptotically
supported by the unstable-neutral subspace only, i.e., it is spanned by the
backward Lyapunov vectors with non-negative exponents. This behavior is at the
core of algorithms known as Assimilation in the Unstable Subspace, although a
formal proof was still missing.
  This paper provides the analytical proof of the convergence of the Kalman
filter covariance matrix onto the unstable-neutral subspace when the dynamics
and the observation operator are linear and when the dynamical model is
error-free, for any, possibly rank-deficient, initial error covariance matrix.
The rate of convergence is provided as well. The derivation is based on an
expression that explicitly relates the error covariances at an arbitrary time
to the initial ones. It is also shown that if the unstable and neutral
directions of the model are sufficiently observed and if the column space of
the initial covariance matrix has a non-zero projection onto all of the forward
Lyapunov vectors associated with the unstable and neutral directions of the
dynamics, the covariance matrix of the Kalman filter collapses onto an
asymptotic sequence which is independent of the initial covariances. Numerical
results are also shown to illustrate and support the theoretical findings.
