Matrices commuting with a given normal tropical matrix
Matrices commuting with a given normal tropical matrix
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DOI：
10.1016/j.laa.2015.04.032
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发表时间：
2012-09
影响因子：
1.1
通讯作者：
J. Linde;M. J. Puente
中科院分区：
文献类型：
--
作者：
J. Linde;M. J. Puente
考虑正方阵X=(X i j) / R∪{−∞}的空间mnnor，即−∞≤X i j≤0且X i i= 0。赋予M n nor热带和⊕和乘法⊙。确定一个实数矩阵a∈M n nor，并考虑M n nor中与a可交换的矩阵的集合Ω (a)，证明Ω (a)是凹形多边形的有限并；特别地，Ω (A)是凸集的有限并。X的集合Ω A (A)使得A⊙X= X⊙A= A也是凹形多面体的有限并。对于X的集合Ω ' (A)也是如此，使得A⊙X= X⊙A= X。则集合Ω A (A)是单位矩阵i的一个邻域。如果A是严格正规的，则Ω ' (A)是零矩阵的一个邻域。在一种情况下，Ω (A)是A的邻域。我们给出了Ω ' (A)维数的上界。我们探讨了当A和X交换时多面体配合物张成A、张成X和张成（A X）之间的关系。两个矩阵，记为A *和A¯，由A产生，与Ω (A)有关。举例说明了它们的几何意义。我们给出了在任何维度上都可以交换的矩阵的例子。
Consider the space M n nor of square normal matrices X=(x i j) over R∪{−∞}, ie,−∞≤ x i j≤ 0 and x i i= 0. Endow M n nor with the tropical sum⊕ and multiplication⊙. Fix a real matrix A∈ M n nor and consider the set Ω (A) of matrices in M n nor which commute with A. We prove that Ω (A) is a finite union of alcoved polytopes; in particular, Ω (A) is a finite union of convex sets. The set Ω A (A) of X such that A⊙ X= X⊙ A= A is also a finite union of alcoved polytopes. The same is true for the set Ω′(A) of X such that A⊙ X= X⊙ A= X. A topology is given to M n nor. Then, the set Ω A (A) is a neighborhood of the identity matrix I. If A is strictly normal, then Ω′(A) is a neighborhood of the zero matrix. In one case, Ω (A) is a neighborhood of A. We give an upper bound for the dimension of Ω′(A). We explore the relationship between the polyhedral complexes span A, span X and span (A X), when A and X commute. Two matrices, denoted A ̲ and A¯, arise from A, in connection with Ω (A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.