Rank-one covariance spike ($\Sigma =I+\theta u{u}^{\top }$)

For a unit vector $u$ and $\theta >0$, a rank-one covariance spike is the covariance matrix $I+\theta u{u}^{\top }$. Its eigenvalue is $1+\theta$ in the direction $u$ and one on the orthogonal complement. It arises by adding an independent scalar normal distribution signal $Yu$ to a standard Gaussian random vector.