I'm interested in the special case of matrices in $\mathrm{SL}_2(\mathbb{C})$. The Frobenius norm of such a matrix $m=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ is $\|m\|:=\sqrt{|a|^2+|b|^2+|c|^2+|d|^2}$. We have from the Cauchy-Schwartz inequality that $\forall m,m': \|mm'\|\leq\|m\|\|m'\|$.
I am interested in a lower bound in terms of $m'$. I'm seeing some notation in other questions involving $\sigma$ but it is not defined. How does that go?
On
Let $A$ and $B$ be matrices with shapes compatible for multiplication. If $b_j$ is the $j$th column of $B$, then $Ab_j$ is the $j$th column of $AB$ and so one computes $$ \|AB\|_F² = \sum_j \|Ab_j\|^2 \ge \sum_j \sigma_\min(A)^2\|b_j\|^2 = \sigma_{\min}(A)^2\|B\|^2_F. $$
We conclude that $\|AB\|_F \ge \sigma_\min(A)\|B\|_F$.
Indeed, you have $\|mm'\|\ge\sigma\|m'\|$, where $\sigma$ is the smallest singular value of $m$. To see this, apply the singular value decomposition of $m$.