Convex Sets of Full Rank Matrices

Abstract

Let A = (a _ij ) and B = (b _ij ) be n-by-m real matrices. Using the notion of a block P-matrix [2] we give a necessary and sufficient condition for the set r(A, B) (c(A, B), resp.) of n-by-m matrices whose rows (columns, resp.) are independent convex combinations of the rows (columns, resp.) of A and B to consist entirely of full row (column, resp.) rank matrices. This improves a result on the set r(A, B) (c(A, B), resp.) proven in [8]. Moreover, we also derive a sufficient condition for the interval of A and B, i.e., for the set i(A, B) of real n-by-m matrices (t _ij a _ij + (1 − t _ij )b _ij ) with t _ij ∈ [0,1] to have the abovementioned rank property.