The authors consider uncertain linear systems where the uncertainties, in addition to being bounded, also satisfy constraints on their phase, In this context, the authors define the "phase-sensitive structured singular value" (PS-SSV) of a matrix and show that sufficient (and sometimes necessary) conditions for stability of such uncertain linear systems can be rewritten as conditions involving PS-SSV. They then derive upper bounds for PS-SSV, computable via convex optimization. They extend these results to the case where the uncertainties are structured (diagonal or block-diagonal, for instance).