Several recent papers demonstrate the interest of viewing systolic algorithms as while-programs whose statements are synchronous multiple assignments. This approach is based on the classical invariant method and compares favourably with earlier ones, based on recurrence systems and space-time transformations. Our purpose is to use the particularities of the systolic paradigm to reduce the creativity needed to develop a systolic algorithm and its invariant. More precisely, two points are taken into account. First, the architecture is often chosen before the real beginning of the development and, second, the basic operations to be executed by individual cells are also partially known at the beginning. In fact, the development does not start from scratch, but from a “generic systolic array” (gsa), whose parameters have to be instantiated. Most systolic arrays are instances of a simple gsa that is introduced, investigated and illustrated in this paper.