Cross-positive matrices revisited
Let \(\Pi(K)\) for a closed, pointed \(n\)-dimensional convex cone \(K\) in \(\mathbb{R}^n\) denote the set of all \(n\times n\) matrices mapping \(K\) into itself, \(\Sigma(K)\) the set of cross-positive matrices, and \(\Lambda\) the set of real multiples of the identity matrix. The authors answer a question of \textit{H. Schneider} and \textit{M. Vidyasagar} [SIAM J. Numer. Anal. 7, 508-519 (1970; Zbl 0245.15008)] by showing that \(\Pi(K)+ \Lambda\neq \Sigma(K)\) for almost all \(K\). They also show a weaker equality of \textit{R. Stern} and \textit{H. Wolkowicz} [SIAM J. Matrix Anal. Appl. 15, 755-778 (1994; Zbl 0806.15007)] fails for almost all \(K\).
scientific article; zbMATH DE number 778260
scientific article