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I would like to prove that for any positive semidefinite matrix $\Sigma_{n\times n}$ and any vector $\mu_{n\times1}$, are there always data points (no matter how many, as long as finite) whose empirical covariance matrix is $\Sigma_{n\times n}$ and the empirical vector is $\mu_{n\times1}$. What is the way to prove it rigorously?

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marked as duplicate by Community Jun 16 '16 at 5:46

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  • $\begingroup$ Feels like it could be shown by giving a way to construct such a sample - easy along one dimension, perhaps not much more difficult with more dimensions? $\endgroup$ – Björn Jun 16 '16 at 5:37
  • $\begingroup$ @josliber and Björn thanks for your comments! I figured out that by Cholesky decomposition, one can easily show $\Simga$ and $\mu$ can be reconstructed by $n+1$ data points. And I will mark this question as duplicated. $\endgroup$ – user112758 Jun 16 '16 at 5:46