COMPLETION FOR TIGHT SIGN-CENTRAL MATRICES

A real matrix A is called a sign-central matrix if for, every matrix $\tilde{A}$ with the same sign pattern as A, the convex hull of columns of $\tilde{A}$ contains the zero vector. A sign-central matrix A is called a tight sign-central matrix if the Hadamard (entrywise) product of any two columns of A contains a negative component. A real vector x = $(x_1,{\ldots},x_n)^T$ is called stable if $\|x_1\|{\leq}\|x_2\|{\leq}{\cdots}{\leq}\|x_n\|$. A tight sign-central matrix is called a $tight^*$ sign-central matrix if each of its columns is stable. In this paper, for a matrix B, we characterize those matrices C such that [B, C] is tight ($tight^*$) sign-central. We also construct the matrix C with smallest number of columns among all matrices C such that [B, C] is $tight^*$ sign-central.