P<inf>†</inf>-matrices: A generalization of P-matrices

For A, B ∈ ℝn×n, 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 and Tsatsomeros have shown that the set r(A,B) (c(A,B)) consists entirely of nonsingular matrices if and only if BA-1 (B-1A) is a P-matrix. For A,B ∈ ℝn×n, let i(A, B) = {C ∈ ℝn×n: min{aij, bij} ≤ cij ≤ max{aij, bij}}. Rohn has shown that if all the matrices in i(A, B) are invertible, then BA-1, A-1B, AB-1 and B-1A are P-matrices. In this article, we define a new class of matrices called P†-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. © 2013 © 2013 Taylor & Francis.