Let $\bf A$ be a $m \times n$ matrix and $\bf B$ be a $n \times k$ matrix. Let $\sigma_{\min} ({\bf M})$ denote the minimum singular value of matrix $\bf M$. Does the following hold?
$$\sigma_{\min} ({\bf AB}) \geq \sigma_{\min} ({\bf A}) \, \sigma_{\min} ({\bf B})$$