Proper Splittings of Rectangular Matrices

Proper splittings $A = M - N$ of a rectangular matrix A of ${\poeratorname{rank}}r$ are characterized in terms of splittings $A_{11} = M_{11} - N_{11} $, where $A_{11} $ is a nonsingular submatrix of A of order r.It is shown that $\rho ( {M^\dag N} ) = \rho ( {M_{11}^{ - 1} \,N_{11} } )$. Conditions for $M^\dag N$ to converge are given, including monotonicity type conditions under regularity assumptions.