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I know that any covariance matrix must be positive semi-definite, but are there any regimes where it is reasonable to assume that covariance matrix is postive definite.

I want to use this to be able to assume that the covariance matrix is invertible, and use the full matrix in a $\chi^{2}$-fit and the Backus-Gilbert method both requiring invertability/positive definite.

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You find a characterization of covariance matrices with zero eigen values here: Sufficient and necessary conditions for zero eigenvalue of a correlation matrix

So you have to ask yourself, whether linear dependencies among the variables may exist.

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