半张量积下可交换的Tropical矩阵

Commutative Semi-Tensor Product under Tropical Matrix

摘要: 本文基于Tropical代数，给出了Tropical代数框架下左(右)半张量积和张量积的定义及其基本性质。随后，证明了两个置换矩阵的张量积仍然是置换矩阵，并将这一结果推广到广义置换矩阵的情形。进一步，利用上述结论，证明了在半张量积下存在可与非奇异幂等矩阵交换的广义置换矩阵。此外，将右半张量积下与非奇异幂等矩阵可交换的广义置换矩阵分块后，得到满足条件的分块矩阵存在与之可交换的矩阵。

Abstract: Based on the tropical algebra, this paper presents the definitions and fundamental properties of the left (right) semi-tensor product and tensor product under the framework of tropical algebra. Subsequently, it is proved that the tensor product of two permutation matrices is still a permutation matrix, and this result is generalized to the case of generalized permutation matrices. Furthermore, by virtue of the above conclusions, it is demonstrated that there exist generalized permutation matrices commuting with non-singular idempotent matrices under the semi-tensor product. In addition, after block decomposition of the generalized permutation matrices that commute with non-singular idempotent matrices under the right semi-tensor product, it is obtained that there exist matrices commuting with the block matrices satisfying the given conditions.