Let's imagine an arbitrary data set in $ D=3 $, is it possible that the covariance matrix consists of diagonal elements only?
I'd say such a data set can't exist. Or would all data points be incident with three axes?
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1$\begingroup$ 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 $\endgroup$– HenryAug 23, 2021 at 13:03
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3$\begingroup$ It could exist a diagonal covariance matrix if variable are uncorrelated. $\endgroup$– Abdoul HakiAug 23, 2021 at 13:05
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$\begingroup$ @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? $\endgroup$– BenAug 23, 2021 at 13:46
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$\begingroup$ @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? $\endgroup$– BenAug 23, 2021 at 13:47
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1$\begingroup$ Try finding the covariances with my suggested eight points (remember to subtract the means). All three will be $0$. Then try to understand why $\endgroup$– HenryAug 23, 2021 at 14:04
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For instance, if we try to estimate linear regression model, we then check an assumption of an absence of autocorrelation (particular, in time series). We use, at first, covariance matrix between residuals with zero non diagonal elements. Then we use Newey West matrix with consistent estimates and compare this matrix to the previous (also, you could use tests to detect).
So, according to the theory, yes, this matrix exists, but in practice, we always get nonzero values (maybe, small, e.g. 1e-5) on non diagonal elements.
