Convex sets of nonsingular and P:–Matrices

Abstract

We show that the set r(A,B) (resp. c(A,B) of square matrices whose rows (resp. columns) are the independent convex combinations of the rows (resp.columns) of real matrices A and B consists entirely of nonsingular matrices if and only if BA −1(resp. B −1 A) is a P-matrix. This imrpoves a theorem on P-Matrices proven in [2] and [3], in the context of interval nonsingularity. We also show that every real P-matrix admits a representation BA −1 with the above property. These reseults are only partially true for complex P-matrices. Based on them we obtain a characterizaiton of complex P-matrices in terms of block partitions.