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arXiv 1209.0660math.RA
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与给定正规热带矩阵可交换的矩阵

Matrices commuting with a given normal tropical matrix

J. Linde, M. J. de la Puente

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摘要：

考虑$\mathbb{R}\cup\{-\infty\}$上的正规方阵$X=(x_{ij})$的空间$M_n^{nor}$，即，$-\infty\le x_{ij}\le0$和$x_{ii}=0$。赋予$M_n^{nor}$热带总和$\oplus$和乘法$\odot$。固定一个实矩阵$A\in M_n^{nor}$，并考虑$M_n^{nor}$中与$A$可交换的矩阵的集合$Ω(A)$。我们证明了$Ω(A)$是一个有限联盟的凹多面体，特别是，$Ω(A)$是一个有限联盟的凸集。$X$的集合$Ω^A(A)$使得$A\odot X=X\odot A=A$也是凹多面体的有限并集。对于$X$的集合$Ω'(A)$也是如此，使得$A\odot X=X\odot A=X$。给出了$M_n^{nor}$的拓扑结构。然后，集合$Ω^{A}(A)$是单位矩阵$I$的邻域。如果$A$是严格正规的，则$Ω'(A)$是零矩阵的邻域。在一种情况下，$Ω(A)$是$A$的邻域。我们给出了$Ω'(A)$维数的一个上界。我们探讨了多面体复合物$span A$，$span X$和$span (AX)$之间的关系，当$A$和$X$交换。两个矩阵，表示为$\underline{A}$和$\bar{A}$，由$A$和$Ω(A)$产生。作为一个例子，详细地给出了它们的几何意义。我们产生的例子矩阵交换，在任何维度。

英文摘要：

Consider the space $M_n^{nor}$ of square normal matrices $X=(x_{ij})$ over $\mathbb{R}\cup\{-\infty\}$, i.e., $-\infty\le x_{ij}\le0$ and $x_{ii}=0$. Endow $M_n^{nor}$ with the tropical sum $\oplus$ and multiplication $\odot$. Fix a real matrix $A\in M_n^{nor}$ 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 $A\odot X=X\odot A=A$ is also a finite union of alcoved polytopes. The same is true for the set $Ω'(A)$ of $X$ such that $A\odot X=X\odot A=X$. 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 $span (AX)$, when $A$ and $X$ commute. Two matrices, denoted $\underline{A}$ and $\bar{A}$, 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.

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