  The main goal of this paper is to characterise all the possible Ces\`aro and
$L$-asymptotic limits of powerbounded, complex matrices. The investigation of
the $L$-asymptotic limit of a powerbounded operator goes back to Sz.-Nagy and
it shows how the orbit of a vector behaves with respect to the powers. It turns
out that the two types of asymptotic limits coincide for every powerbounded
matrix and a special case is connected to the description of the products
$SS^*$ where $S$ runs through those invertible matrices which have unit
columnvectors. We also show that for any powerbounded operator acting on an
arbitrary complex Hilbert space the norm of the $L$-asymptotic limit is greater
than or equal to 1, unless it is zero; moreover, the same is true for the
Ces\`aro asymptotic limit of a not necessarily powerbounded operator, if it
exists.
