Is the minimum singular value (amplitude) of $\boldsymbol{AB}$ bounded by the counterparts of $\boldsymbol{A}$ and $\boldsymbol{B}$?

Question

Consider two complex matrices $\boldsymbol{A} \in \mathbb{C}^{M\times N}$ and $\boldsymbol{B} \in \mathbb{C}^{N\times Q}$. Is the minimum singular value's amplitude of $\boldsymbol{AB}$ bounded by the minimum singular value amplitudes of $\boldsymbol{A}$ and $\boldsymbol{B}$? That is,

does $$ |\sigma_{min}(\boldsymbol{AB})| \leq \min \{|\sigma_{min}(\boldsymbol{A})|,|\sigma_{min}(\boldsymbol{B})|\} $$ hold?

Answer 1

No, it doesn't even work for scalar matrices.

An inequality of this type should be homogenous, of the correct degree.

The inequality you do have is $$\sigma_{\min}(A B) \ge \sigma_{\min}(A) \cdot \sigma_{\min}(B)$$