Rump, Siegfried M.Siegfried M.Rump2021-05-052021-05-052000-06-20Linear Algebra and Its Applications 278 (1-3): 121-132 (1998)http://hdl.handle.net/11420/9440For 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.en0024-379520001-3121132American Elsevier Publ.Condition numberStructured perturbationsSymmetric matricesMathematikJournal Article10.1016/S0024-3795(97)10078-7Journal Article