condition number after scaling matrix Maybe a well-known question. Let $\Sigma$ represent a real symmetric positive definite matrix, i.e. a covariance matrix.
Which diagonal matrix $D$ with positive diagonal minimizes the condition number $\frac{\lambda_{\max}(D\Sigma D)}{\lambda_{\min}(D\Sigma D)}$?
In contrast to some other literature on preconditioners, the pre- and post multiply matrix $D$ are the same here. There could be a clue here if the pre- en post-D would be allowed to differ. Unfortunately, I have no access to eq. (A.74).
Numerical experiments suggest that the identity matrix is always optimal if $\Sigma$ is two dimensional. For larger dimensions, the optimum is nontrivial.

There is no simple relation (at least nowadays) for the optimal diagonal scaling which minimizes the spectral condition number of a matrix except for several special cases.
The classical result for SPD matrices is due to van der Sluis, which states that for an SPD matrix $A\in\mathbb{R}^{n\times n}$,
$$
\kappa_2(DAD)\leq n\min_{\text{$\tilde{D}\in\mathbb{R}^{n\times n}$ diagonal}}\kappa_2(\tilde{D}A\tilde{D}),
$$
where $D=\mathrm{diag}(a_{11}^{-1/2},\ldots,a_{nn}^{-1/2})$.
There is, however, a paper by Braatz and Morari from 1994 where the authors propose a formulation of the optimal scaling problem leading to a convex optimization problem.
