= Compound-symmetry covariance
{title2=$\Sigma=aI+cJ$}
A <compound-symmetry covariance> matrix has a common diagonal entry and a common off-diagonal entry: $\Sigma=aI+cJ$. For a group of size $k$, its within-group <eigenvalue> is $a$ and its constant-direction <eigenvalue> is $a+kc$. Thus $a\geq0$ and $a+kc\geq0$ characterize <positive semidefiniteness>. When $c\geq0$, an independent shared <random intercept> plus individual noise realizes this covariance. A group mean has <variance> $c+a/k$; a difference of two disjoint subgroup means within the same group cancels $c$.
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