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Nowadays, certain problems in automata theory, manufacturing systems, telecommunication networks, parallel processing systems and traffic control are intimately linked with linear systems over tropical semirings. Due to non-invertibility of matrices—except monomial matrices—over certain semirings, we cannot generally take advantage of having the inverse of the coefficient matrix of a system to solve it. The main purpose of this paper is to introduce two methods based on the pseudo-inverse of a matrix for solving a linear system of equations over tropical semirings. To this end, under suitable conditions, we first reduce the order of the system through some row–column operational analysis. We then present a necessary and sufficient condition for the system to have a maximal solution. This solution is also obtained through a new version of Cramer’s rule for overdetermined system of equations. Finally, some illustrative examples are given to show the efficiency of the proposed methods, and Maple procedures are also included in the end.

In this paper, we discuss 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 go over 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. Importantly, we show that the maximal solutions obtained from these methods as well as the previously studied LU-method are all identical.

In this paper, we introduce and analyze a new LU-factorization technique for square matrices over idempotent semifields. In particular, more emphasis is put on “max-plus” algebra here but the work is extended to other idempotent semifields as well. We first determine the conditions under which a square matrix has LU factors. Next, using this technique, we propose a method for solving square linear systems of equations whose system matrices are LU-factorizable. We also give conditions for an LU-factorizable system to have solutions. This work is an extension of similar techniques over fields. Maple® procedures for this LU-factorization are also included.

In this paper, we introduce and analyze a normalization method for solving a system of linear equations over tropical semirings. We use a normalization method to construct an associated normalized matrix, which gives a technique for solving the system. If solutions exist, the method can also determine the degrees of freedom of the system. Moreover, we present a procedure to determine the column rank and the row rank of a matrix. Flowcharts for this normalization method and its applications are included as well.