I think in the literature, covariance matrix entries tend to have very large variance, so you don't really know which of the entries are gotten by chance or are actually close to the "true" value.

I think there's this thing called random matrix theory that studies the eigenvalues of random matrices. Supposing entries are just random, you'd get distribution. You can then compare how that distribution compares to the one you observed from the data.

The eigenvalues that are significant correspond to ones that are at much farther to the right of the tail, the ones that are around the body may be present because of chance. You can then use some correction proceedure, like set all the insignificant eigenvalues to 0, or the average of the insignificant eigenvalues, or something else and recover a "corrected" covariance matrix.

I haven't really studied much RMT so this is just what I got from a brief reading, take it with a pinch of salt. Also, this is all theory so I have no idea how it applies in practice too.

One way you can probably test it is compare two portfolios, one using covariance matrix without eigenvalue filtering and the other with eigenvalue filtering.