On the Maximal Solution of A Linear System over Tropical Semirings

Jamshidvand, Sedighe; Ghalandarzadeh, Shaban; Amiraslani, Amirhossein; Olia, Fateme

doi:10.48550/arxiv.1904.13169

Cited by 2 publications

(4 citation statements)

References 9 publications

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“…Now, we introduce a new version of Cramer's rule to obtain the maximal solution of a linear system, based on the pseudo-inverse of the system matrix. In the next theorem, we present the extended Cramer's rule over idempotent semifields whenever the ε-determinant of the system matrix is nonzero, which is an extension of Theorem 4 in [3]. Proof.…”

Section: Methods For Solving Linear Systems Over Idempotent Semifieldsmentioning

confidence: 99%

“…First we introduce the pseudo-inverse method to obtain a maximal solution of the linear system. In the following theorem, we present a necessary and sufficient condition on the system matrix over idempotent semifields which is an extension of Theorem 3 in [3] for "max −plus algebra ".…”

Section: Methods For Solving Linear Systems Over Idempotent Semifieldsmentioning

confidence: 99%

“…This makes linear system solution over idempotent semifields ever so important. In this work, we intend to extend the solution methods presented in [3] and [7], namely, the pseudo-inverse method, the extended Cramer's rule, and the normalization method, to idempotent semifields.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of linear systems over idempotent semifields

Olia¹,

Ghalandarzadeh²,

Amiraslani³

et al. 2019

Preprint

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In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule which is also related to the pseudo-inverse of the system matrix. In these two methods, the constant vector plays an implicit role in solvability of the system. Another method is called the normalization method in which both the system matrix and the constant vector play explicit roles in the solution process. Each of these methods yields the maximal solution if it exists. Finally, we show the maximal solutions obtained from these methods and some previous methods are all identical.

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Section: Methods For Solving Linear Systems Over Idempotent Semifieldsmentioning

confidence: 99%

Section: Methods For Solving Linear Systems Over Idempotent Semifieldsmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of linear systems over idempotent semifields

Olia¹,

Ghalandarzadeh²,

Amiraslani³

et al. 2019

Preprint

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“…It doesn't necessarily have an inverse to the ⊗ operation, so matrix 𝐴 is an invertible matrix if it meets certain conditions. One of the isomorphic tropical semirings with max-plus is a min-plus algebra (Jamshidvand et al, 2019).…”

Section: A Introductionmentioning

confidence: 99%

Determining the Inverse of a Matrix over Min-Plus Algebra

Siswanto,

Gusmizain

2024

JTAM

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Linear algebra over the semiring R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' ) is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i<ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.

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On the Maximal Solution of A Linear System over Tropical Semirings

Cited by 2 publications

References 9 publications

Analysis of linear systems over idempotent semifields

Analysis of linear systems over idempotent semifields

Determining the Inverse of a Matrix over Min-Plus Algebra

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