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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 $r$.

The given equicorrelation covariance matrix can be writtenOn $1=(1,1,1,1{)}^T$, the rank-one matrix $11^T$ acts as multiplication by four. On the three-dimensional orthogonal complement $\{v:{\sum }_j{v}_j=0\}$ it acts as zero. Hence the eigenvalues are $1+3r$ and $1-r$ with multiplicity three.

For $0<r<1$, the largest eigenvalue is $1+3r$ and its normalized eigenvector is $1/2$. Therefore the first population principal component isOmitting the centering merely changes the component by a constant and has no effect on its variance. Since $\mathrm{tr}\Sigma =4$, its explained fraction isIt rises from one quarter towards one as the positive common correlation increases. Outside the specified range, a valid covariance requires $-1/3\le r\le 1$; for negative $r$ the sum direction is no longer the leading one.