P†-matrices: a generalization of P-matrices

For A, B is an element of R-nxn, let r(A,B) (c(A,B)) be the set of matrices whose rows, (columns) are independent convex combinations of the rows (columns) of A and B. Johnson andTsatsomeros have shown that the set r(Lambda,B) (c(Lambda,B)) consists entirely of nonsingular matrices if and only if BA-1(B-1 A) is a P-matrix. For A, B is an element of R-nxn, let i (A, B) = {C is an element of R-nxn : min{a(i) (j), b(i) (j)} <= c(i) (j) <= max{a(i) (j), b(i) (j)}}. Rohn has shown that if all the matrices in i (A, B) are invertible, then BA(-1), A(-1)B, AB(-1) and B-1 A are P-matrices. In this article, we define a new class of matrices called P-dagger-matrices and present certain extensions of the above results to the singular case, where the usual inverse is replaced by the Moore-Penrose generalized inverse. The case of the group inverse is briefly discussed.