Since $$Z=X^\top (XX^\top + \mathrm{Id})^{-1} X=(Y + \mathrm{Id})^{-1} Y,$$ with $Y=X^\top X$, we can find the eigenvalue density of $Z$ from the eigenvalue density of $Y$, which for large matrix dimensions is given by the Marchenko-Pastur density $\rho_{\rm MP}(y)$.

In particular, the average eigenvalue is given by $$\langle z\rangle=\int_0^\infty \frac{y}{1+y}\,\rho_{\rm MP}(y)\,dy=\frac{\sqrt{\gamma^2+4}+\gamma+2}{2 \gamma}.$$

Since $$Z=X^\top (XX^\top + \mathrm{Id})^{-1} X=(Y + \mathrm{Id})^{-1} Y,$$ with $Y=X^\top X$, we can find the eigenvalue density of $Z$ from the eigenvalue density of $Y$, which for large matrix dimensions is given by the Marchenko-Pastur density $\rho_{\rm MP}(y)$.

In particular, if the matrix elements of $X$ have variance $1/m$, the average $\langle z\rangle$ of an eigenvalue of $Z$ is given by $$\langle z\rangle=\int_0^\infty \frac{y}{1+y}\,\rho_{\rm MP}(y)\,dy=\frac{\sqrt{\gamma^2+4}+\gamma+2}{2 \gamma},$$ in the limit $n,m\rightarrow\infty$ at constant $\gamma=n/m\in(0,1]$.

Since $$Z=X^\top (XX^\top + \mathrm{Id})^{-1} X=(Y + \mathrm{Id})^{-1} Y,$$ with $Y=X^\top X$, we can find the eigenvalue density of $Z$ from the eigenvalue density of $Y$, which for large matrix dimensions is given by the Marchenko-Pastur formula.

Since $$Z=X^\top (XX^\top + \mathrm{Id})^{-1} X=(Y + \mathrm{Id})^{-1} Y,$$ with $Y=X^\top X$, we can find the eigenvalue density of $Z$ from the eigenvalue density of $Y$, which for large matrix dimensions is given by the Marchenko-Pastur density $\rho_{\rm MP}(y)$.

In particular, the average eigenvalue is given by $$\langle z\rangle=\int_0^\infty \frac{y}{1+y}\,\rho_{\rm MP}(y)\,dy=\frac{\sqrt{\gamma^2+4}+\gamma+2}{2 \gamma}.$$