  Consider the planar linear switched system $\dot
x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times2$ real
matrices, $x \in \R^2$, and $u(.):[0,\infty[\to\{0,1\}$ is a measurable
function. In this paper we consider the problem of finding a
(coordinate-invariant) necessary and sufficient condition on $A$ and $B$ under
which the system is asymptotically stable for arbitrary switching functions
$u(.)$. This problem was solved in previous works under the assumption that
both $A$ and $B$ are diagonalizable. In this paper we conclude this study, by
providing a necessary and sufficient condition for asymptotic stability in the
case in which $A$ and/or $B$ are not diagonalizable. To this purpose we build
suitable normal forms for $A$ and $B$ containing coordinate invariant
parameters. A necessary and sufficient condition is then found without looking
for a common Lyapunov function but using "worst-trajectory'' type arguments.
