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. Motivated by the latter, 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 largest and smallest. Moreover, unlike the latter classical kappa condition number, omega is not invariant under inversion, an important point that allows one to recall that it is the conditioning of the inverse that is important. Our study is in the context of optimal conditioning for: (i) low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) iterative methods for linear systems; (iia) clustering of eigenvalues; (iib) convergence rates; and (iic) estimating the actual condition of a linear system. We emphasize that the simple functions in omega allow one to exploit optimality conditions and derive explicit formulae for omega-optimal preconditioners of special structure. Connections to partial Cholesky type sparse preconditioners are made that modify the iterates of Cholesky decomposition by including the entire diagonal at each iteration. Our results confirm the efficacy of using the omega-condition number compared to the classical kappa-condition number.