Suppose $\bf W$ is a $n \times n$ matrix with every diagonal entry is $0$. Additionally, the row sum and column sum can be also uniformly bounded by a constant as $n\xrightarrow{} \infty$. Assume $|\rho|<1$ and $\bf I$ refers to the identity matrix. How can I lower-bound the smallest eigenvalue of the following positive semidefinite matrix?
$$({\bf I} -\rho {\bf W})^\top ({\bf I} -\rho {\bf W})$$
From my numerical experiments, I guess this result mainly depends on $\rho$ and is irrelevant to $n$. If $\rho$ is near $1$, the lower bound will be near $0$. While $\rho$ is near $0$, the lower bound would be much larger.