Structured Perturbations Part I: Normwise Distances

In this paper we study the condition number of linear systems, the condition number of matrix inversion, and the distance to the nearest singular matrix, all problems with respect to normwise structured perturbations. The structures under investigation are symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel matrices (some results on other structures such as tridiagonal and tridiagonal Toeplitz matrices, both symmetric and general, are presented as well). We show that for a given matrix the worst case structured condition number for all right-hand sides is equal to the unstructured condition number. For a specific right-hand side we give various explicit formulas and estimations for the condition numbers for linear systems, especially for the ratio of the condition numbers with respect to structured and unstructured perturbations. Moreover, the condition number of matrix inversion is shown to be the same for structured and unstructured perturbations, and the same is proved for the distance to the nearest singular matrix. It follows a generalization of the classical Eckart–Young theorem, namely, that the reciprocal of the condition number is equal to the distance to the nearest singular matrix for all structured perturbations mentioned above.