Geodesic
The Moore-Penrose inverse of a free matrix
The electronic journal of linear algebra, Tome 16 (2007), pp. 208-215.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 05C50, 15A09
Keywords: free matrix, Moore-Penrose inverse, bipartite graph, directed graph 
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Britz, Thomas. The Moore-Penrose inverse of a free matrix. The electronic journal of linear algebra, Tome 16 (2007), pp. 208-215. http://geodesic.mathdoc.fr/item/ELA_2007__16__a19/