The solution sets of max-algebraic linear equation systems

Article type: Research Article

Authors: Li, Shi-ju^{a; *} | Wang, Hui-li^{b; †}

Affiliations: [a] College of Fine Arts, Sichuan Normal University, Chengdu, Sichuan, People’s Republic of China | [b] School of Science, Southwest Petroleum University, Chengdu, Sichuan, People’s Republic of China

Correspondence: [*] Corresponding author. Shi-ju Li, College of Fine Arts, Sichuan Normal University, Chengdu, Sichuan 610101, People’s Republic of China. E-mails: lishiju@sicnu.edu.cn; 784760668@qq.com.

Correspondence: [† ] E-mail: 201499010027@swpu.edu.cn

Abstract: This paper deals with the max-algebraic linear equation system A ⊗ x = b. As in the conventional linear algebra such a linear system may have none, exactly one or infinitely many solutions. When the number of solutions is exactly one, Cramer’s rule is given as an analogue of the classical linear algebra. When the number of solutions is infinite, the existence of a minimal solution is shown and the formula of minimal solution is given. Furthermore, it is proved that every solution can be expressed as a linear combination of a respective minimal solution and some special vectors. Finally, an algorithm to describe all the solutions of a given max-algebraic linear equation system is proposed when its number of solutions is infinite. AMS classification: 15A80,15A06

Keywords: Max-algebra, linear equation system, solution set, minimal solution, cramer’s rule

DOI: 10.3233/JIFS-182911

Journal: Journal of Intelligent & Fuzzy Systems, vol. 37, no. 4, pp. 5105-5111, 2019