Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g., in quasi-Newton methods. We study a nonclassic matrix condition number, the \(\omega \) -condition number, \(\omega \) for short. \(\omega \) is the ratio of: the arithmetic and geometric means of the singular values, rather than the largest and smallest for the classical \(\kappa \) -condition number. The simple functions in \(\omega \) allow one to exploit first order optimality conditions. We use this fact to derive explicit formulae for (i) \(\omega \) -optimal low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) \(\omega \) -optimal preconditioners of special structure for iterative methods for linear systems. In the latter context, we analyze the benefits of \(\omega \) for (a) improving the clustering of eigenvalues; (b) reducing the number of iterations; and (c) estimating the actual condition of a linear system. Moreover we show strong theoretical connections between the \(\omega \) -optimal preconditioners and incomplete Cholesky factorizations, and highlight the misleading effects arising from the inverse invariance of \(\kappa \) . Our results confirm the efficacy of using the \(\omega \) -condition number compared to the \(\kappa \) -condition number.
