Given a regular rational matrix function A(z) we describe the set of vector-valued functions g(z) of the form \(g(z)=A(z)f(z)\) for a vector- valued function f(z) which is analytic at a prescribed point \(z_ 0\). In the scalar case, the solution is quite simple, involving the order of the pole or zero of A(z) at \(z_ 0\). The matrix case is complicated when \(z_ 0\) is both a pole and a zero for \(z_ 0\). In addition to a ''zero pair'' and a ''pole pair'' for A(z) at \(z_ 0\), a new invariant called the ''coupling matrix'' is needed for a complete description. The inverse problem of deciding when a ''data set'' arises from a matrix function in this way is also solved. As an application, the authors show how to obtain state space formulas for the linear fractional map which parametrizes all solutions of a Nevanlinna-Pick interpolation problem.