Let $\|\cdot\|_F$ be the Frobenius norm on $M_{n\times n}$ and $\|\cdot\|_2$ the Euclidean norm on $\mathbb{R}^n$. For a given matrix $A\in M_{n\times n}$, what vectors satisfy $\|Av\|_2 = \|A\|_F\|v\|_2$?
I can prove the relation is $\leq$ as follows
$$ \|Av\|_2^2 = \sum\limits_i\left(\sum\limits_j A_{ij}v_j\right)^2 = \sum\limits_i\left(\left<A_i,v\right>\right)^2 \leq \sum\limits_i\|A_i\|_2^2\|v\|_2^2 = \|v\|_2^2 \sum\limits_i\sum\limits_j A_{ij}^2=\|v\|_2^2\|A\|_F^2 $$ where $A_i$ is the $i$-th row of $A$, $\left<\cdot,\cdot\right>$ is the standard inner product, and the inequality is due to the Cauchy-Schwartz inequality.
Cauchy Schwartz inequality is an equality if and only if the vectors are linearly dependent. Thus, in order to get an equality, we must have for all $i$, $A_i = \alpha_i v$ for some $\alpha_i\in\mathbb{R}$.
Does this mean that we get an equality if and only if $(i)$ $\det(A)=0$ and $(ii)$ $v$ is a row of $A$? This is a very strict condition. Is it the only way, or am I missing something?