The size of the smallest structured destabilizing perturbation for a linear time-invariant system can be calculated via the structured singular value (mu). The function mu can be bounded above by the solution of a convex optimization problem, and in general there is a gap between mu and the convex bound, This paper gives an alternative characterization of mu which is used to study this gap for low-rank matrices. The ion-rank characterization provides an easily computed bound which can potentially be significantly better than the standard convex bound. This is used to find new examples with larger gaps than previously known.