ELA, Volume 16, pp. 208-215, August 2007, abstract.

The Moore-Penrose Inverse of a Free Matrix

Thomas Britz

A matrix is free, or generic, if its nonzero entries are 
algebraically independent. Necessary and sufficient combinatorial 
conditions are presented for a complex free matrix to have a free 
Moore-Penrose inverse. These conditions extend previously known 
results for square, nonsingular free matrices. The result used to 
prove this characterization relates the combinatorial structure 
of a free matrix to that of its Moore-Penrose inverse. Also, it is 
proved that the bipartite graph or, equivalently, the zero pattern
of a free matrix uniquely determines that of its Moore-Penrose inverse, 
and this mapping is described explicitly. Finally, it is proved that a 
free matrix contains at most as many nonzero entries as does its 
Moore-Penrose inverse.