Efficiency of GLS over OLS, proving through semi-definite and positive semi-definite

JHi all! Just browsing through some materials on proving efficiency of GLS over OLS, vice versa, and one of the methodology was to compare their respective variances, that for OLS to be more efficient, the difference between both, GLS - OLS variance should be positive semi-definite.

Therefore, when there's an actual numerical or algebraic question, does one look to check if the difference between both variance is greater than 0 ? (to explain if it may be positive definite or positive semi-definite)

How to definite if that (the difference of two variance) is positive definite or positive semi-definite?

Separately there seem to be a vector expression evaluation for potential positive definite / semi-definite which am not sure if may be related...

Thank you!

• If you know the error covariance matrix, GLS is always at least as efficient as OLS, with the equality only occurring if the error covariance matrix $\Sigma = \sigma^2I$. You don't need to check, it's always true. Sep 12 at 16:40
• Hello! Thanks for coming back :-) May i just check if there are any resources to determining if the difference is then positive semi-definite or positive definite? Or the question will ask you to prove either one and it's not something that has to be deduced from the answer whether the difference is p.s.d or p.d, thanks! Sep 13 at 0:15
• It's not clear to me what your question actually is. Are you looking for a proof that GLS is at least as efficient as OLS? The difference is always positive semi-definite (with positive definite as a subset of PSD.) Otherwise, we wouldn't be able to state that GLS was more efficient than OLS. Sep 13 at 2:12