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A linear vector equation in two unknown vectors is examined in the frameworkof tropical algebra dealing with the theory and applications of semirings andsemifields with idempotent addition. We consider a two-sided equation whereeach side is a tropical product of a given matrix by one of the unknownvectors. We use a matrix sparsification technique to reduce the equation to aset of vector inequalities that involve row-monomial matrices obtained from thegiven matrices. An existence condition of solutions for the inequalities isestablished, and a direct representation of the solutions is derived in acompact vector form. To illustrate the proposed approach and to compare theobtained result with that of an existing solution procedure, we apply oursolution technique to handle two-sided equations known in the literature.Finally, a computational scheme based on the approach to derive all solutionsof the two-sided equation is discussed.

Using matrix sparsification to solve tropical linear vector equations