Systems of linear partial differential equations with constant coefficients, like their ordinary differential equation counterparts, can be characterized by the properties of the matrices that form the coefficients of the differential operators. The question arises: Do the matrix operators that result from partial differential equations possess eigenvalues and eigensolutions in the same way that ordinary differential matrix operators do? The answer to this question is explored in some detail using as an example the linearized flow of a viscous fluid. It is shown that eigenfactors do exist for these equations, and that, of necessity, these involve hypercomplex algebra. This fact introduces significant new features to the problem. It is shown that eigenmodes exist and that each of these has its distinctive fundamental singularity. The fluid mechanical significance of these is examined in some detail. In addition, a representative group of other partial differential equations is examined and their eigenmodes and fundamental singularities are determined. It is shown that a number of basic differences exist between the eigenfunction theory for ordinary and for partial differential equations.