After having tried this with randomly generated vectors, I am consistently seeing the correlation matrix of randomly generated numbers, regardless of which distribution they are sampled from, are always more well-conditioned than the covariance matrix. Which is strange because the covariance matrix exists before the correlation matrix: the correlation matrix must be computed from the covariance matrix, and the other way around cannot be done.
In other words, the covariance matrix, being more ill-conditioned, in fact is transformed into a more well-conditioned, stable, matrix when it is converted to the correlation matrix.
which makes me wonder if all the financial models that rely on the covariance matrix would be better of using the correlation matrix as an input instead, given all the animosity towards the instability and ill-conditioning of the covariance. I know that the covariance possess variance, or risk, so slanting models to strictly interpret correlations instead would result in missing out on the more relevant measure, which is risk, not correlation, so it seems that we are putting interpretability first compared to other highly-related options, which comes at the price of numerical instability and estimation error