  The Ensemble Kalman inversion (EnKI), proposed by Iglesias et al. for the
solution of Bayesian inverse problems of type $y=Au+\varepsilon$, with $u$
being an unknown parameter and $y$ a given datum, is a powerful tool usually
derived from a sequential Monte Carlo point of view. It describes the dynamics
of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical
measure is sampled from the prior, evolving over an artificial time $t$ towards
an approximate solution of the inverse problem.
  Using spectral techniques, we provide a complete description of the dynamics
of EnKI and their asymptotic behavior in parameter space. In particular, we
debunk a common working assumption in the field, saying that in the linear
Gaussian regime, the empirical measure at time $t=1$ is a surrogate for the
posterior, i.e., the correct Bayesian answer to the inverse problem.
Furthermore, we show that residuals in parameter space do not decrease
monotonously in the Euclidean norm and suggest a problem-adapted norm, where
monotonicity can be proved. Finally, we derive a system of ordinary
differential equations governing the spectrum and eigenvectors of the
covariance matrix.
