Upper bound for smallest eigenvalue for the matrix $\mathbf{S} (\mathbf{S}^{T} \mathbf{S} + \lambda \mathbf{A})^{-2} \mathbf{S}^{T}$

Question

My book claims that the smallest eigenvalue of the matrix

$$\mathbf{S} (\mathbf{S}^{T} \mathbf{S} + \lambda \mathbf{A})^{-2} \mathbf{S}^{T},$$

where $\mathbf{S}$ is an $n \times k $ matrix and $\mathbf{A}$ is a $k \times k$ positive semi-definite matrix and $\lambda>0$, is proportional to $1/\lambda$. Could you please provide an argument for this? It's not immediately obvious to me.

Thank you in advance.