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.