Solution of generalized linear vector equations in idempotent algebra

The problem on the solutions of homogeneous and nonhomogeneous generalized linear vector equations in idempotent algebra is considered. For the study of equations, an idempotent analog of matrix determinant is introduced and its properties are investigated. In the case of irreducible matrix, existence conditions are found and the general solutions of equations are obtained. The results are extended to the case of arbitrary matrix. As a consequence the solutions of homogeneous and nonhomogeneous inequalities are presented. 1. Introduction. For analysis of different technical, economical, and engineering systems the problems are often occurred which require the solution of vector equations linear in a certain idempotent algebra [1–5]. As a basic object of idempotent algebra one usually regards a commutative semiring with an idempotent summation, a zero, and a unity. At once many practical problems give rise to idempotent semiring, in which any nonzero (in the sense of idempotent algebra) element has the inverse one by multiplication. Taking into account a group property of multiplications, such a semiring are called sometimes idempotent semifield. Note that in passing from idempotent semrings to semifields, the idempotent algebra takes up an important common property with a usual linear algebra. In this case it is naturally expected that the solution of certain problems of idempotent algebra can be obtained by a more simple way and in a more conventional form, in particular, due to the applications of idempotent analogs of notions and results of usual algebra. Consider, for example, the problem on the solution with respect to the unknown vector x the equation A ⊗x ⊕b = x, where A is a certain matrix, b is a vector, ⊕ and ⊗ are the signs of operations of summation and multiplication of algebra. Different approaches to the solution of this equation were happily developed in the work [3–7] and the others. However many of these works consider a general case of idempotent semiring and, therefore, the represented in them results have often too general theoretical nature and are not always convenient for practical application. In a number of works it is mainly considered existence conditions of solution of equations and only some its partial (for example, minimal) solution is suggested in explicit form. In the present work a new method for the solution of linear equations in the case of idempotent semiring with the inverse one by multiplication (a semifield) is suggested which can be used for obtaining the results in compact form convenient for as their realization in the form of computational procedures as a formal analysis. For the proof of certain assertions the approaches, developed in [1, 2, 4], are used. In the work there is given first a short review of certain basic notions of idempotent algebra [2, 4, 5, 8], involving the generalized linear vector spaces and the elements of matrix calculus, and a number of auxiliary