On structured singular values

The author shows how to estimate the norm of the smallest block-structured additive perturbation of a block 2×2 matrix that makes it singular. The estimates are accurate to within a factor of at most 3^3/2 (a factor of three for real matrices) and work for all possible block perturbation of a block 2×2 matrix and for a large class of matrix norms, including all p-norms and the Frobenius norm. Using an algorithm of W. Hager (1984) the author estimates bounds even for large matrices. He explicitly exhibits rank one or rank two perturbations which achieve his upper bounds. These explicit perturbations can be used as starting values for an optimization routine designed to compute the answer to higher accuracy than the present a priori estimates provide. These results extend to some block perturbations of 3×3 block matrices, although the upper and lower bounds may not always be close