The aim of this work is to develop analogue concepts of column reduced polynomial matrices for proper rational matrices. A definition of column reducedness for a class of proper rational matrices is proposed and the properties of such matrices are studied, in particular reduction to column reduced form by elementary operations over the ring of proper rational functions, and the relationship between the degrees of the invariant factors of a column reduced matrix and the so-defined column indices. The physical significance of such matrices in terms of their finite structure is explained; this interpretation completely complements the physical interpretation of a column reduced polynomial matrix. An application of the properties of column reduced proper rational matrices to the decoupling problem is also presented: the infinite structure which can be obtained while decoupling a linear multivariable system by non-regular static state feedback is completely characterized.