  A central question in verification is characterizing when a system has
invariants of a certain form, and then synthesizing them. We say a system has a
$k$ linear invariant, $k$-LI in short, if it has a conjunction of $k$ linear
(non-strict) inequalities -- equivalently, an intersection of $k$ (closed) half
spaces -- as an invariant. We present a sufficient condition -- solely in terms
of eigenvalues of the $A$-matrix -- for an $n$-dimensional linear dynamical
system to have a $k$-LI. Our proof of sufficiency is constructive, and we get a
procedure that computes a $k$-LI if the condition holds. We also present a
necessary condition, together with many example linear systems where either the
sufficient condition, or the necessary is tight, and which show that the gap
between the conditions is not easy to overcome. In practice, the gap implies
that using our procedure, we synthesize $k$-LI for a larger value of $k$ than
what might be necessary. Our result enables analysis of continuous and hybrid
systems with linear dynamics in their modes solely using reasoning in the
theory of linear arithmetic (polygons), without needing reasoning over
nonlinear arithmetic (ellipsoids).
