Schneider, Hans; Vidyasagar, Mathukumalli
Cross-positive matrices.
SIAM J. Numer. Anal. 7 1970 508--519.
15.30 (65.00)
Let $C$ be a cone in Euclidean $n$-space $R^n$, $C^0$ its interior and $C^*$ its polar cone, i.e., $C^*=\{z\in R^n\colon(z,y)\geq 0,y\in C\}$. An $n\times n$-matrix $A$ is said to be "cross-positive on $C$" if, for all $y\in C$ and $z\in C^*$ with $(y,z)=0$, one has $(z,Ay)\geq 0$, "strongly cross-positive on $C$" if $A$ is cross-positive on $C$ and, for each $y\in C (y\neq 0)$, there is a $z\in C$ such that $(z,y)=0$ and $(z,Ay)>0$, "strictly cross-positive on $C$" if, for all $y\neq 0$ and $z\neq 0$ with $(z,y)=0$, one has $(z,Ay)>0$. The matrix $A$ is said to be irreducible on $C$ if $AC\subseteq C$, $A$ has no eigenvector in $\partial C$ and $(I+A)^{n-1}(C\backslash\{0\})\subseteq C^0$. Theorems are given on relations between the sets of matrices satisfying one or the other of these conditions. It is shown that $A$ is cross-positive on $C$ if and only if $B_t=\exp(tA)$ is positive on $C$, i.e., $B_tC\subseteq C$ for all $t\geq 0$. The matrix $B_t$ is irreducible on $C$ for all $t>0$, except possibly a countable set, if $A$ is strongly cross-positive. There follows a discussion of the relation between these definitions and theorems on the one hand, and the Perron-Frobenius theorem for cones on the other hand. The above theorems are then made more specific for polyhedral cones $C$, i.e., cones with a finite set of generators. Finally the results extending the Perron-Frobenius theorem are strengthened for the case of symmetric matrices. The paper ends with a set of 5 open questions.
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