Some consequences of the rank normal form of a matrix

If A is a rectangular matrix of rank r, then A may be written as PSQ where P and Q are invertible matrices and s=(IrOOO)s = \left( {\matrix{ \hfill {{{\rm{I}}_{\rm{r}}}} & \hfill {\rm{O}} \cr \hfill {\rm{O}} & \hfill {\rm{O}} \cr } } \right). This is the rank normal form of the matrix A.

The purpose of this paper is to exhibit some consequences of this representation form.