COMPLETION FOR TIGHT SIGN-CENTRAL MATRICES

Abstract

A real matrix A is called a sign-central matrix if for, every matrix <TEX>$\tilde{A}$</TEX> with the same sign pattern as A, the convex hull of columns of <TEX>$\tilde{A}$</TEX> 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 = <TEX>$(x_1,{\ldots},x_n)^T$</TEX> is called stable if <TEX>$\|x_1\|{\leq}\|x_2\|{\leq}{\cdots}{\leq}\|x_n\|$</TEX>. A tight sign-central matrix is called a <TEX>$tight^*$</TEX> 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 (<TEX>$tight^*$</TEX>) sign-central. We also construct the matrix C with smallest number of columns among all matrices C such that [B, C] is <TEX>$tight^*$</TEX> sign-central.