# What weights should be used when adjusting a correlation matrix to be positive definite?

I have a correlation matrix $A$ for an equity market that is not positive definite. Higham (2002) proposes the Alternating Projections Method, minimising the weighted Frobenius norm $||A-X||_W$ where $X$ is the resulting positive definite matrix.

How should one choose the weight matrix $W$?

The easy alternative is to weigh them equally (W is an identity matrix), but if one has exposures to a portfolio, wouldn't it be natural to weigh the correlations according to your weights of exposure in the different assets, in order to alter their historical correlation less than for those assets you have little exposure in? Or is there a more natural choice?

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Hi Osloguten, welcome to quant.SE and thanks for submitting this very relevant question. – Tal Fishman Oct 26 '11 at 18:41
Thanks. Well, so far I have not found any solution and are currently running unweighted approximations. I find this about alright, but as I am approximating correlations from some stocks that are somewhat illiquid it would be satisfying knowing that these will be altered more than the main stocks in our portfolios.. – AdAbsurdum Nov 8 '11 at 8:18