Matrix arithmetic is extensively used in the field of Modern Control Theory, as it is the most efficient way of solving the equations that describe the behavior of complex systems. Systolic algorithms have proved as an effective way to perform matrix calculation, currently enforced by the use of FPGAs for their implementation. To avoid the requirement of a specific design and implementation for each problem, we propose the construction of a Systolic Library that can be easily used to build the solution for each concrete equation. This paper presents the main ideas taken into account when designing the Library, and shows its application to the solution of the triangular case of the Sylvester Equation.