Finding commutative multidimensional downsamplers and upsamplers

A systematic approach to solve the open problem of finding multidimensional upsamplers and downsamplers that satisfy commutative conditions is derived. A test for conversion commutivity of conventional television signals to high definition television signals requires two matrices to be commutative and coprime, but this does not give a discovery process like the one presented. The solution is an algorithm in four steps.