Let $S$ be a random $M\times N$ matrix with independently identically distributed entries. The Pastur-Marcenko law gives the spectral density of $S^T S$ as $N\rightarrow\infty$ with a fixed ratio $\alpha=M/N$:

$$\rho_{\textrm{MP}} (\lambda) = \left\{\begin{array}{ll} \frac{\alpha}{2 \pi \lambda \sigma^2} \sqrt{(b - \lambda) (\lambda - a)} + (1 - \alpha) \delta (\lambda) & a < \lambda < b\\ 0 & \textrm{otherwise} \end{array}\right.$$

where $a,b=\left( 1 \pm \sqrt{\alpha} \right)^2 \sigma^2 / \alpha$.

Let $e_1$ be the Cartesian unit vector in a fixed direction and consider the rank-one updated matrix $A = S^T S + e_1e_1^T$. The spectral density of $A$ has an analytical formula in the large $N$ limit?


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