In this paper, we present a parametrization of piecewise linear (PWL) Lyapunov functions. To this end, we consider the class of all continuous PWL functions defined over a simplicial partition. We take advantage of a recently developed high level canonical PWL (HL CPWL) representation, which expresses the PWL function in a compact and closed form. Once the parametrization problem is properly stated, we focus on its application to the stability analysis of dynamic systems. We consider uncertain non-linear systems and extend the sector condition obtained by Ohta ct al. In addition, we propose a method of selecting an optimal candidate. One of the main advantages of this approach is that the parametrization and choice of the Lyapunov candidate, as well as the stability analysis, result in linear programming problems.