An optimization problem, which arises in various applications as that of minimizing the span seminorm, is considered in the framework of tropical mathematics. The problem is to minimize a nonlinear function defined on vectors over an idempotent semifield, and calculated by means of multiplicative conjugate transposition. We find the minimum of the function, and give a partial solution which explicitly represents a subset of solution vectors. We characterize all solutions by a system of simultaneous equation and inequality, and exploit this characterization to investigate properties of the solutions. A matrix sparsification technique is developed to extend the partial solution to a wider solution subset, and then to a complete solution described as a family of subsets. We offer a backtracking procedure that generates all members of the family, and derive an explicit representation for the complete solution. Numerical examples and graphical illustrations of the results are presented.