论文标题
在矩阵的类别上，排从对角线弱降低了循环的行
On the class of matrices with rows that weakly decrease cyclicly from the diagonal
论文作者
论文摘要
我们考虑满足$ a_ {ij {ii} \ geq a_ {i，i+1} \ geq \ geq \ geq a__ {in} 1，\ dots，n $。使用这样的矩阵$ a $，我们将有向图$ g（a）$关联。我们证明了系统$ a^t x =λe$的解决方案，其中$λ\ in \ mathbb {r} $和$ e $是所有$ a $ a $ a^vector的$ a^t x = e $的“基本”解决方案的线性组合，并与$ \ ker a^t $相关联的$ contectove $ \ ker a^t $ contect $ concontect $ contect（c）（c。这使我们可以根据封闭的SCC数量和$ a^t x = e $的解决方案来表征$ \ det a $的标志。此外，我们为$ a $提供条件为$ p $ -matrix。
We consider $n\times n$ real-valued matrices $A = (a_{ij})$ satisfying $a_{ii} \geq a_{i,i+1} \geq \dots \geq a_{in} \geq a_{i1} \geq \dots \geq a_{i,i-1}$ for $i = 1,\dots,n$. With such a matrix $A$ we associate a directed graph $G(A)$. We prove that the solutions to the system $A^T x = λe$, with $λ\in \mathbb{R}$ and $e$ the vector of all ones, are linear combinations of 'fundamental' solutions to $A^T x=e$ and vectors in $\ker A^T$, each of which is associated with a closed strongly connected component (SCC) of $G(A)$. This allows us to characterize the sign of $\det A$ in terms of the number of closed SCCs and the solutions to $A^T x = e$. In addition, we provide conditions for $A$ to be a $P$-matrix.
