Karamardian Matrices：$Q$ -矩阵的推广
Karamardian Matrices: A Generalization of $Q$-Matrices
摘要：
如果线性互补问题$LCP(A,q)$对所有$q \in \mathbb{R}^n$都有解，则称实方阵$A$为$Q$ -矩阵.这意味着对于每个载体$q$，存在载体$x$，使得$x \geq 0, y=Ax+q\geq 0$和$x^Ty=0$。Karamelo的一个著名结果表明，如果问题$LCP(A,0)$和$LCP(A,d)$对于某些$d\in \mathbb{R}^n, d >0$只有零解，则$A$是一个$Q$ -矩阵.通过放松对$d$的条件，并对上述两个问题的解向量$x$施加一个条件，作者引入了一类新的矩阵，称为Karamartim矩阵，要求这两个修正问题只有零解.在这篇文章中，系统的处理Karamanet矩阵进行。除其他事项外，它示出了如何Karamartists矩阵的属性是类似于那些$Q$ -矩阵。最近引入的概念的$P_{\#}$ -矩阵的一个子类具有Karamanovic属性，并为此原因，我们进行了深入的研究$P_{\#}$ -矩阵，并作出一些基本的贡献。
英文摘要：
A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem $LCP(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$ and $x^Ty=0$. A well known result of Karamardian states that if the problems $LCP(A,0)$ and $LCP(A,d)$ for some $d\in \mathbb{R}^n, d >0$ have only the zero solution, then $A$ is a $Q$-matrix. By relaxing the condition on $d$ and imposing a condition on the solution vector $x$ in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, requiring that these two modified problems have only zero as a solution. In this article, a systematic treatment of Karamardian matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of $Q$-matrices. A subclass of a recently introduced notion of $P_{\#}$-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of $P_{\#}$-matrices and make some fundamental contributions.
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