A convergent algorithm for finding a minimum norm real matrix perturbation that reduces the rank of a general complex matrix

The problem of computing a real matrix perturbation having a minimum norm which causes a general complex matrix to drop rank is examined. Given the state model describing a linear time-invariant system, the norm of this matrix perturbation helps to determine the robustness of several system properties with respect to real parameter variations. The norm of such a perturbation is known to be a discontinuous function in the space of complex matrices. Aspects of the continuity of the problem are reviewed, and a convergent algorithm is presented. The algorithm computes a sequence of real matrix perturbations. A cluster point of this sequence of matrix perturbations satisfies the necessary condition for being a minimum norm, rank-reducing perturbation for some matrix that is arbitrarily close to the given complex matrix