Geodesic
Generalized inverses of bordered matrices
The electronic journal of linear algebra, Tome 10 (2003), pp. 16-30.

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

Summary: Several authors have considered nonsingular borderings A = s, B CD X t^ of B and investigated the properties of submatrices of A-1. Under specific conditions on the bordering, one can recover any g-inverse of B as a submatrix of A-1. Borderings A of B are considered, where A might be singular, or even rectangular. If A is m * n and if B is an r * s submatrix of A, the consequences of the equality m + n - $rank(A) = r + s - rank(B)$ with reference to the g-inverses of A are studied. It is shown that under this condition many properties enjoyed by nonsingular borderings have analogs for singular (or rectangular) borderings as well. We also consider g-inverses of the bordered matrix when certain rank additivity conditions are satisfied. It is shown that any g-inverse of B can be realized as a submatrix of a suitable g-inverse of A, under certain conditions.
Classification : 15A09, 15A03
Keywords: generalized inverse, Moore-Penrose inverse, bordered matrix, rank additivity 
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     title = {Generalized inverses of bordered matrices},
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Bapat, R.B.; Zheng, Bing. Generalized inverses of bordered matrices. The electronic journal of linear algebra, Tome 10 (2003), pp. 16-30. http://geodesic.mathdoc.fr/item/ELA_2003__10__a23/