This may be trivial.
While covariance matrices of random variables are positive semi-definite, does the converse hold true as well, that positive semi-definite matrices are also valid covariance matrices of random variables?
Wikipedia says this is the case, however, I don't follow the argument:
... the covariance matrix of a multivariate probability distribution is always positive semi-definite. (...) Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.
Grateful for any explanations.