# Covariance matrix with diagonal elements only

Let's imagine an arbitrary data set in $$D=3$$, is it possible that the covariance matrix consists of diagonal elements only?

I'd say such a data set can't exist. Or would all data points be incident with three axes? • Take the eight points $(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)$ as your and find the covariance matrix Aug 23, 2021 at 13:03
• It could exist a diagonal covariance matrix if variable are uncorrelated. Aug 23, 2021 at 13:05
• @Henry The data points (0,1,1), (1,0,1), (1,1,0) and (1,1,1) would result in a S with non-diagonal elements, or not?
– Ben
Aug 23, 2021 at 13:46
• @AbdoulHaki Sure, I just wonder about the possibility of such a data set. Therefore the question: Only when all the data points would be exactly positioned on the axes, or not?
– Ben
Aug 23, 2021 at 13:47
• Try finding the covariances with my suggested eight points (remember to subtract the means). All three will be $0$. Then try to understand why Aug 23, 2021 at 14:04