# Eigenvalue density of $S^TS + ee^T$

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?