arXiv 1401.01342014-01math.OC

余正矩阵的极小零点

Minimal zeros of copositive matrices

Roland Hildebrand

备注：: Some conditions and proofs simplified

摘要：

设$A$是共正锥${\cal C}_n$的一个元素。$A$的零$u$是一个非零非负向量，使得$u^TAu = 0$。$u$的支持是对应于$u$的正条目的索引集$\mbox{supp}u \subset \{1,\dots,n\}$。如果不存在$A$的另一个零$v$使得它的支集$\mbox{supp}v$是$\mbox{supp}u$的严格子集，则称$A$的零$u$是极小的。研究了余正矩阵的极小零点的性质及其支撑。特别注意的是，对于半正定矩阵的锥$S_+(n)$是不可约的余正矩阵，即，矩阵不能写为一个正和一个非零半正定矩阵的和。给出了矩阵$A$关于$S_+(n)$按其极小零点不可约的一个充要条件。对于不可约矩阵关于按项非负矩阵的锥${\cal N}_n$给出了一个类似的条件。对于$n = 5$矩阵，关于$S_+(5)$和${\cal N}_5$都不可约的矩阵是极值的。对于$n = 6$，提供了例外极值矩阵可以具有的最小零点的支持的候选组合的列表。

英文摘要：

Let $A$ be an element of the copositive cone ${\cal C}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set $\mbox{supp}u \subset \{1,\dots,n\}$ corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ such that its support $\mbox{supp}v$ is a strict subset of $\mbox{supp}u$. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone $S_+(n)$ of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone ${\cal N}_n$ of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and ${\cal N}_5$ are extremal. For $n = 6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.

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摘要：

英文摘要：

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