ELA, Volume 10, pp. 16-30, January 2003, abstract.
Generalized inverses of bordered matrices
Ravindra B. Bapat and Bing Zheng
Several authors have considered nonsingular
borderings
A = [ B C ]
[ D X ]
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 mxn and if B is an rxs
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.