Equicorrelation covariance matrix ($\Sigma =(1-r)I_p+r11^T$)

This unit-variance covariance matrix has eigenvalue $1+(p-1)r$ on the constant-vector direction and eigenvalue $1-r$ on its orthogonal complement. It is positive semidefinite exactly when $-1/(p-1)\le r\le 1$. For positive $r$, the first population principal component is the normalized sum of the centered variables.