Is every symmetric positive semi-definite matrix a covariance of some multivariate distribution?

Question

One can easily prove that every [covariance matrix](http://en.wikipedia.org/wiki/Covariance_matrix) is [positive semi-definite](http://en.wikipedia.org/wiki/Positive-definite_matrix). I come across many claims that the converse is also true; that is, *Every symmetric positive semi-definite matrix is a covariance marix of some multivariate distribution.* Is it true? If it is, how can we prove it?

Answer 1

Hello,

Yes, every symmetric, positive semi-definite matrix is the covariance matrix of some multivariate distribution. To prove this fact, it suffices to give an example. Let Σ be an nxn positive semi-definite matrix, and let μ be any n-dimensional real vector. Then we can simply take the (n-dimensional) multivariate normal distribution with mean μ and covariance matrix Σ as the desired example. [If an n-dimensional random vector X has the multivariate normal distribution mentioned here, we often designate this by X∼N(μ,Σ).]

Hope that helps,

William