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.