I have a sample covariance matrix that is non positive-semi definite (due to missing data points). I am looking at a number of techniques to 'fix' my covariance matrix and make it positive semi-definite so that I can use PCA, Markowitz portfolio optimisation, etc.

I was wondering, is there a way to check the quality of my correction to the covariance matrix? You'd want to keep as much of the original covariance as possible, but at the same time ensure it was positive-semi definite. Is there some potential metric based on that?

Any ideas are well appreciated.