An exact solution for a special class of cone-preserving linear matrix inequalities (LMIs) is developed. By using a generalized version of the classical Perron-Frobenius theorem, the optimal value is shown to be equal to the spectral radius of an associated linear operator. This allows For a much more efficient computation of the optimal solution using, for instance, power iteration-type algorithms. This particular LMI class appears in the computation of upper bounds for some generalizations of the structured singular value mu (spherical mu) and in a class of rank minimization problems previously studied, Examples and comparisons with existing techniques are provided.