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. We do this
in the context of optimal conditioning for: 
(i) our application to low rank updating of generalized Jacobians;
(ii) iterative methods for linear systems: 
(iia) clustering of eigenvalues and (iib) convergence rates.

For a positive definite matrix, the $\omega$-condition measure is 
the ratio of the arithmetic and
geometric means of the eigenvalues. In particular, our applications
concentrate on linear systems with low rank updates of ill-conditioned 
positive definite matrices.
These systems arise in the context of nonsmooth Newton methods using
generalized Jacobians. We are able to use optimality conditions
and derive \emph{explicit} formulae for $\omega$-optimal preconditioners and
preconditioned updates. Connections to partial Cholesky
sparse preconditioners are made.

Evaluating or estimating the classical condition number $\kappa$ can be
expensive. We show that the $\omega$-condition number can be evaluated
explicitly following a Cholesky or LU factorization. Moreover, the
simplicity of $\omega$ allows for the derivation of formulae for optimal
preconditioning in various scenarios, i.e.,~this avoids the need for
expensive algorithmic calculations. And, our empirics
show that $\omega$ estimates the
actual condition of a linear system significantly better. Moreover,
our empirical results show a significant decrease in the number of
iterations required for a requested accuracy in the residual during an
iterative method, i.e.,~these results confirm the efficacy of using the 
$\omega$-condition number compared to the classical condition number.

Finally, we discuss the fact that the condition number of the
$\kappa$-condition number is the $\kappa$-condition number but this is
not true for the $\omega$-condition number further illustrating the
improved conditioning using the latter condition number.