  We discuss computational procedures based on descriptor state-space
realizations to compute proper range space bases of rational matrices. The main
computation is the orthogonal reduction of the system matrix pencil to a
special Kronecker-like form, which allows to extract a full column rank factor,
whose columns form a proper rational basis of the range space. The computation
of several types of bases can be easily accommodated, such as minimum-degree
bases, stable inner minimum-degree bases, etc. Several straightforward
applications of the range space basis computation are discussed, such as, the
computation of full rank factorizations, normalized coprime factorizations,
pseudo-inverses, and inner-outer factorizations.
