论文标题
Karamardian矩阵：$ Q $ - 久的概括
Karamardian Matrices: A Generalization of $Q$-Matrices
论文作者
论文摘要
如果线性互补问题$ lcp（a，q）$具有所有$ q \ in \ mathbb {r}^n $的解决方案。这意味着每个向量$ q $都存在一个向量$ x $，因此$ x \ geq 0，y = ax+q \ geq 0 $和$ x^ty = 0 $。 Karamardian的一个众所周知的结果指出，如果问题$ lcp（a，0）$和$ lcp（a，d）$，对于某些$ d \ in \ mathbb {r}^n，d> 0 $，则只有零解决方案，那么$ a $ a $ as $ as $ q $ -mmatrix。通过在$ d $上放松条件，并在上述两个问题中对解决方案向量$ x $施加条件，作者引入了一种新的称为karamardian矩阵的矩阵，要求这两个修改后的问题仅作为解决方案为零。在本文中，进行了卡拉马德矩阵的系统处理。除其他外，它显示了Karamardian矩阵的属性类似于$ q $ - amatrices的属性。最近引入的$ p _ {\＃} $ - 矩阵的一个子类显示具有Karamardian财产，因此，我们对$ p _ {\＃} $ - 矩阵进行了详尽的研究，并做出了一些基本的贡献。
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.
