= Rank of a centered sample covariance matrix
{title2=$\operatorname{rank}S=\operatorname{rank}(HX)\leq\min(p,n-1)$}
If $H=I_n-\mathbf1\mathbf1^T/n$ and $S=X^THX/n$, then $\operatorname{rank}S=\operatorname{rank}(HX)\leq\min(p,n-1)$. In particular <full column rank> of raw $X$ does not ensure a nonsingular centered <sample covariance matrix>. When $S$ is singular, its Gaussian precision <profile likelihood> has no finite maximizer: increase precision along a nonzero null vector.
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