= Equicorrelation covariance matrix
{title2=$\Sigma=(1-r)I_p+r\mathbf1\mathbf1^T$}
For $p\geq2$ variables with unit <variances> and common pairwise <correlation> $r$, the <equicorrelation covariance matrix> is
$$
\Sigma=(1-r)I_p+r\mathbf1\mathbf1^T.
$$
On the constant-vector direction, $\mathbf1\mathbf1^T$ 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 to
$$
-\frac1{p-1}\leq r\leq1.
$$
At $r=1$, 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 $r$.
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