锥上的线性方程与Collatz-Wielandt数
Linear Equations over cones and Collatz-Wielandt numbers
摘要：
设$K$是$\IR^n$中的真锥，$A$是满足$AK\subseteq K$的$n\times n$实矩阵，$b$是$K$的给定向量，$λ$是给定正实数.本文考虑下列两个线性方程：（i）$(λI_n-A)x=b$，$x\in K$，（ii）$(A-λI_n)x=b$，$x\in K$。我们得到了第一类方程可解的几个等价条件。对于第二个方程，我们给出了一个等价的条件，它的可解性的情况下，当$λ>ρ_b (A)$，我们也发现了一个必要条件时，$λ=ρ_b (A)$，也当$λ< ρ_b(A)$，充分接近$ρ_b(A)$，其中$ρ_b (A)$表示的局部谱半径的$A$在$b$。在$λ$固定的情况下，我们还考虑了集合$(A-λI_n)K \bigcap K$何时等于$\{0\}$或$K$的问题，以及由集合生成的$K$的面是什么。然后，我们得到了与保锥映射相关的局部谱半径和Collatz-Wielandt集（或数）的一些新结果，并推广了$Z$ -矩阵中$M$-矩阵的一个已知的交替序列刻画.
英文摘要：
Let $K$ be a proper cone in $\IR^n$, let $A$ be an $n\times n$ real matrix that satisfies $AK\subseteq K$, let $b$ be a given vector of $K$, and let $λ$ be a given positive real number. The following two linear equations are considered in this paper: (i)$(λI_n-A)x=b$, $x\in K$, and (ii)$(A-λI_n)x=b$, $x\in K$. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when $λ>ρ_b (A)$, and we also find a necessary condition when $λ=ρ_b (A)$ and also when $λ< ρ_b(A)$, sufficiently close to $ρ_b(A)$, where $ρ_b (A)$ denotes the local spectral radius of $A$ at $b$. With $λ$ fixed, we also consider the questions of when the set $(A-λI_n)K \bigcap K$ equals $\{0\}$ or $K$, and what the face of $K$ generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of $M$-matrices among $Z$-matrices in terms of alternating sequences.
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