 Equicorrelation covariance matrix

For $p\ge 2$ variables with unit variances and common pairwise correlation $r$, the equicorrelation covariance matrix isOn the constant-vector direction, $11^T$ has eigenvalue $p$; on its orthogonal complement, it has eigenvalue zero. Hence the eigenvalues of $\Sigma$ are $1+(p-1)r$ and $1-r$, the latter with multiplicity $p-1$. Positive semidefiniteness is therefore equivalent toAt $r=1$, the matrix has matrix rank one. At $r=-1/(p-1)$, the constant direction has zero variance and the matrix has matrix rank $p-1$. In the interior it is positive-definite. For $r>0$, the first population principal component is the normalized sum of the centered variables; for $r<0$, the largest eigenvalue instead occurs on the orthogonal complement.

The dimension restriction matters. For $p=1$, the formula gives $\Sigma =[1]$ for every $r$, so the lower bound involving $p-1$ does not apply: there is no distinct variable pair whose correlation could determine $r$.