Determinant of a matrix over min-plus algebra

Siswanto, Siswanto; Gusmizain, A.; Wibowo, Supriyadi

doi:10.1080/09720529.2021.1948663

Cited by 4 publications

(5 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In min-plus algebra, the determinant of a matrix over min-plus algebra is defined in two approaches called dominant and permanent [14]. Dominant of 𝐴 ∈ ℝ 𝑚𝑖𝑛 𝑛×𝑛 according to Siswanto [14] is…”

Section: Resultsmentioning

confidence: 99%

“…Also, in 2021, Diena [13] discussed about solving a system of linear equation in min-plus algebra using a reduction and discrepancy matrix. In the same year, Siswanto et al [14] have researched about the determinant of a matrix over min-plus algebra. Similarly, in max-plus algebra, the determinant of a matrix over min-plus algebra has no direct analogy to conventional algebra.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cramer’s Rule in Min-Plus Algebra

Putri,

Siswanto,

Kurniawan

2024

BAREKENG: J. Math. & App.

View full text Add to dashboard Cite

Cramer’s rule is one of a method for solving a system of linear equations in conventional algebra. The system of linear equation can be solved using Cramer’s rule if . Max-plus algebra is a set where is a set of real numbers, equipped with biner operations and where and . Min-plus Algebra is a set where is a set of real numbers, equipped with biner operations and where and . In max-plus algebra has been formulated Cramer’s rule to solve a system of linear equations. Because max-plus algebra is isomorphic to min-plus algebra, Cramer’s rule can be formulated into min-plus algebra. The purpose of this research is to determine the sufficient conditions for a system of linear equations can be solved using Cramer’s rule. The method used in this research is a literature study that reviews previous research related to min-plus algebra, max-plus algebra, and Cramer’s rule in max-plus algebra. By using the appropriate analogy in max-plus algebra, we can determine the sufficient conditions so that a system of linear equations in min-plus algebra can be solved using Cramer’s rule. Based on the research, the sufficient conditions for a system of linear equations can be solved using Cramer’s rule are for and with the Cramer’s rule is . For an invertible matrix A, Cramer’s rule can be written as .

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cramer’s Rule in Min-Plus Algebra

Putri,

Siswanto,

Kurniawan

2024

BAREKENG: J. Math. & App.

View full text Add to dashboard Cite

show abstract

“…This study aims to expand the research on minplus algebra based on research in linear algebra and max-plus algebra, and also determine the connections or direct analogues between them. The research begins by defining a polynomial form in min-plus algebra, then determining the characteristic min-polynomial referring to the definition of permanent according to Siswanto, Gusmizain, et al (Siswanto, Gusmizain, et al, 2021). The next step will be determining the eigenvalue of matrices over min-plus algebra through its characteristic min-polynomial.…”

Section: B Methodsmentioning

confidence: 99%

“…𝑘 times Definition 6. (Siswanto, Gusmizain, et al, 2021) For a matrix 𝐴 ∈ ℝ 𝜀 ′ 𝑛×𝑛 the permanent of 𝐴 is defined as…”

Section: Given 𝐴 ∈ ℝ 𝜀 ′mentioning

confidence: 99%

Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra

Maghribi,

Siswanto,

Sutrima

2023

JTAM

View full text Add to dashboard Cite

Let R_ε=R∪{-∞}, with R being a set of all real numbers. The algebraic structure (R_ε,⊕,⊗) is called max-plus algebra. The task of finding the eigenvalue and eigenvector is called the eigenproblem. There are several methods developed to solve the eigenproblem of A∈R_ε^(n×n), one of them is by using the characteristic max-polynomial. There is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is a set of R_(ε^' )=R∪{+∞} that uses minimum (⊕^' ) and addition (⊗) operations. The eigenproblem in min-plus algebra is to determine λ∈R_(ε^' ) and v∈R_(ε^')^n such that A⊗v=λ⊗v. In this paper, we provide a method for determining the characteristic min-polynomial and solving the eigenproblem by using the characteristic min-polynomial. We show that the characteristic min-polynomial of A∈R_(ε^')^(n×n) is the permanent of I⊗x⊕^' A, the smallest corner of χ_A (x) is the principal eigenvalue (λ(A)), and the columns of A_λ^+ with zero diagonal elements are eigenvectors corresponding to the principal eigenvalue.

show abstract

“…The set of matrices of dimensions 𝑚 row and 𝑛 column wich have entries in ℝ 𝜀 ′ , is known as min-plus algebra matrices and is denoted by ℝ 𝜀 ′ 𝑚×𝑛 (Siswanto, Gusmizain, et al, 2021) f. If 𝐴 ∈ ℝ 𝜀 ′ 𝑛×𝑛 and 𝑘 ∈ ℕ, the 𝑘 power of 𝐴 is determined by…”

Section: Matrix Over Min-plus Algebramentioning

confidence: 99%

Determining the Inverse of a Matrix over Min-Plus Algebra

Siswanto,

Gusmizain

2024

JTAM

View full text Add to dashboard Cite

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.

show abstract

Determinant of a matrix over min-plus algebra

Cited by 4 publications

References 1 publication

Cramer’s Rule in Min-Plus Algebra

Cramer’s Rule in Min-Plus Algebra

Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra

Determining the Inverse of a Matrix over Min-Plus Algebra

Contact Info

Product

Resources

About