We consider linear operators on finite-dimensional semimodules over idempotent semifields (tropical linear operators) and examine extremal properties of their spectrum. It turns out that the minimum value for some nonlinear functionals that involve tropical linear operators and use multiplicative conjugate transposition is determined by their maximum eigenvalue, where the minimum and maximum are thought in the sense of the order induced by the idempotent addition in the carrier semifield. We exploit the extremal properties to solve multidimensional optimization problems formulated in the tropical mathematics setting as to minimize the functionals under constraints in the form of linear equations and inequalities. Complete closed-form solutions are given by using a compact vector representation. Applications of the results to real-world problems are presented, including solutions to both unconstrained and constrained multidimensional minimax single facility location problems with rectilinear and Chebyshev distances.