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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.00384 |
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| _version_ | 1866913491610238976 |
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| author | Krivulin, Nikolai |
| author_facet | Krivulin, Nikolai |
| contents | A linear vector equation in two unknown vectors is examined in the framework of tropical algebra dealing with the theory and applications of semirings and semifields with idempotent addition. We consider a two-sided equation where each side is a tropical product of a given matrix by one of the unknown vectors. We use a matrix sparsification technique to reduce the equation to a set of vector inequalities that involve row-monomial matrices obtained from the given matrices. An existence condition of solutions for the inequalities is established, and a direct representation of the solutions is derived in a compact vector form. To illustrate the proposed approach and to compare the obtained result with that of an existing solution procedure, we apply our solution technique to handle two-sided equations known in the literature. Finally, a computational scheme based on the approach to derive all solutions of the two-sided equation is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00384 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Using matrix sparsification to solve tropical linear vector equations Krivulin, Nikolai Commutative Algebra Systems and Control 15A80 (Primary), 15A06 (Secondary) A linear vector equation in two unknown vectors is examined in the framework of tropical algebra dealing with the theory and applications of semirings and semifields with idempotent addition. We consider a two-sided equation where each side is a tropical product of a given matrix by one of the unknown vectors. We use a matrix sparsification technique to reduce the equation to a set of vector inequalities that involve row-monomial matrices obtained from the given matrices. An existence condition of solutions for the inequalities is established, and a direct representation of the solutions is derived in a compact vector form. To illustrate the proposed approach and to compare the obtained result with that of an existing solution procedure, we apply our solution technique to handle two-sided equations known in the literature. Finally, a computational scheme based on the approach to derive all solutions of the two-sided equation is discussed. |
| title | Using matrix sparsification to solve tropical linear vector equations |
| topic | Commutative Algebra Systems and Control 15A80 (Primary), 15A06 (Secondary) |
| url | https://arxiv.org/abs/2210.00384 |