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 ω-condition number, ω for short. ω is the ratio of: the arithmetic and geometric means of the singular values, rather than the largest and smallest for the classical κ-condition number. The simple functions in ω allow one to exploit first order optimality conditions. We use this fact to derive explicit formulae for (i) ω-optimal low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) ω-optimal preconditioners of special structure for iterative methods for linear systems. In the latter context, we analyze the benefits of ω 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 ω-optimal preconditioners and incomplete Cholesky factorizations, and highlight the misleading effects arising from the inverse invariance of κ. Our results confirm the efficacy of using the ω-condition number compared to the κ-condition number.