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For a system matrix $M$ , this paper explores the smallest (Frobenius) norm additive structured perturbation $\delta M$ for which a system property $\mathcal {P}$ (e.g., controllability, observability, stability, etc.) fails to hold, i.e., $\delta M$ is the structured perturbation with smallest Frobenius norm such that there exists a property matrix $R\in \mathcal {P}$ for which $M{-}\delta M{-}R$ drops rank. The Frobenius norm is used because of its direct dependence on the magnitude of each entry in the perturbation matrix. Necessary conditions on a locally minimum norm structured rank-reducing perturbation $\delta M$ and associated property matrix $R$ are set forth and proven. An iterative algorithm is also set forth that computes a locally minimum norm structured perturbation and associated property matrix satisfying the necessary conditions. Algorithm convergence is proven using a discrete Lyapunov function. [ABSTRACT FROM AUTHOR]