When do several linear operators share an invariant cone
When do several linear operators share an invariant cone
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多个线性算子何时共享一个不变锥体
DOI：
10.1016/j.laa.2010.04.006
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发表时间：
2010
影响因子：
1.1
通讯作者：
V. Protasov
中科院分区：
文献类型：
--
作者：
V. Protasov
我们建立了有限矩阵族具有公不变锥的一个判据。该判据将不变量锥的存在性问题简化为依赖于该族的两个特殊数的相等问题。尽管理论上很简单，但实际使用该标准可能很困难。我们证明了4个具有积分项的矩阵的公不变锥的存在性问题在算法上是不可判定的。给出了该判据的若干推论，并给出了充分必要条件。最后，我们引入了几个矩阵的“共向数”。我们证明了当存在一个矩阵的小扰动时，这个参数接近于零，在此扰动之后，它们得到一个不变锥。给出了一种计算算法。
We establish a criterion for a finite family of matrices to possess a common invariant cone. The criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family. In spite of theoretical simplicity, the practical use of the criterion may be difficult. We show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable. Corollaries of the criterion, which give sufficient and necessary conditions, are derived. Finally, we introduce a “co-directional number” of several matrices. We prove that this parameter is close to zero iff there is a small perturbation of matrices, after which they get an invariant cone. An algorithm for its computation is presented.