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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.