On parallel summability of matrices (Q1073867)

The authors use generalized inverses to characterize the parallel summability (p.s.) of matrices. If A and B are rectangular matrices of the same size, and if there exists C such that \(\{C^-\}=\{A^-+B^- \}\), then A and B are p.s. and \(C=P(A,B)\). (Here \(C^-\) denotes a weak generalized inverse of C and \(P(A,B)=A(A+B)^-B\) is the parallel sum of A and B.) A pair of almost positive definite (a.p.d.) matrices of the same type are p.s. and the parallel sum is a.p.d. of the same types as the summands. Finally, two ''saturated'' subclasses of a.p.d. matrices are identified.