热带矩阵对偶与格林D关系
Tropical matrix duality and Green's D relation
摘要：
本文给出了n阶热带矩阵乘法半群的格林D关系的完整刻画。我们的主要工具是一个新的变体之间的对偶行和列空间的热带矩阵（研究科恩，Gaubert和Quadrat和分别由Cherin和Sturmfels）。与已有的对偶定理不同，我们的版本允许一个逆，从而给出了两个热带凸集是矩阵的行和列空间的一个充要条件。我们还证明了矩阵对偶映射在任意热带矩阵的射影行空间和射影列空间之间诱导出一个等距（关于Hilbert射影度量），并建立了关于Green的其它关系的一些基本结果.
英文摘要：
We give a complete description of Green's D relation for the multiplicative semigroup of all n-by-n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen, Gaubert and Quadrat and separately by Develin and Sturmfels). Unlike the existing duality theorems, our version admits a converse, and hence gives a necessary and sufficient condition for two tropical convex sets to be the row and column space of a matrix. We also show that the matrix duality map induces an isometry (with respect to the Hilbert projective metric) between the projective row space and projective column space of any tropical matrix, and establish some foundational results about Green's other relations.
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