Bull. Korean Math. Soc. 2006 Vol. 43, No. 2, 343-352 Published online June 1, 2006

Myung-Sook Cho and Suk-Geun Hwang Kyungpook National University, Kyungpook National University

Abstract : 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 $\bdx=(x_1,\ldots,x_n)^T$ is called stable if $|x_1| \leq |x_2| \leq \cdots \leq |x_n|$. A tight sign-central matrix is called a {\it{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.