Structured perturbations and symmetric matrices Original Research Article

For a given n × n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal Bauer-Skeel condition number cannot be larger than (3 + 2√2)n. In this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation, for relative perturbations of the individual matrix components. It is shown that a symmetric linear system can be arbitrarily ill-conditioned, while any symmetric and entrywise relative perturbation of the matrix of less than 100% does not produce a singular matrix. That means that the inverse of the condition number and the distance to the nearest ill-posed problem can be arbitrarily far apart. Finally we prove that restricting structured perturbations to symmetric (entrywise) perturbations cannot change the condition number by more than a factor (3 + 2√2)n.