  We consider bounded operators $A$ acting iteratively on a finite set of
vectors $\{f_i : i\in I\}$ in a Hilbert space $\mathcal H$ and address the
problem of providing necessary and sufficient conditions for the collection of
iterates $\{A^nf_i : i\in I,\,n=0,1,2,\ldots\}$ to form a frame for the space
$\mathcal H$. For normal operators $A$ we completely solve the problem by
proving a characterization theorem. Our proof incorporates techniques from
different areas of mathematics, such as operator theory, spectral theory,
harmonic analysis, and complex analysis in the unit disk. In the second part of
the paper we drop the strong condition on $A$ to be normal. Despite this quite
general setting, we are able to prove a characterization which allows to infer
many strong necessary conditions on the operator $A$. For example, $A$ needs to
be similar to a contraction of a very special kind. We also prove a
characterization theorem for the finite-dimensional case. --- These results
provide a theoretical solution to the so-called Dynamical Sampling problem
where a signal $f$ that is evolving in time through iterates of an operator $A$
is spatially sub-sampled at various times and one seeks to reconstruct the
signal $f$ from these spatial-temporal samples.
