Structured Rank-Reducing Matrix Perturbations: Theory and Computation

The paper investigates the problem of computing structured matrix perturbations that cause some specified system matrix to fail to have full rank. It also considers a related problem: finding a perturbation that most nearly causes some specified system matrix to have full rank. The paper discusses theoretical issues concerning the existance of solutions to these problems. It suggests a new approach for problems requiring the differentiation of singular values. Finally an algorithm for finding structured rank-reducing perturbations and structured nearly rank-reducing perturbations are developed. The paper demonstrates convergence of the algorithm to a rank-reducing perturbation or to a local minimum for a nearly rank-reducing perturbation. Numerical examples illustrating the technique are included.