Frobenius norm of rank-constrained matrix product is bounded

Question

Say I have three matrices $\mathbf{W} \in \mathbb{R}^{p \times m}$ and $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}$ with $\operatorname{rank}(\mathbf{A}) \leq r$ and $\mathbf{B}$ is unconstrained on rank. I also assume that all of them are bounded in Frobenius norm by some number say $\Vert\mathbf{W}\Vert_F, \Vert\mathbf{A}\Vert_F, \Vert\mathbf{B}\Vert_F \leq \kappa$

I am trying to show that $ \sup \Vert\mathbf{WA} \Vert_F \leq \sup \Vert\mathbf{WB}\Vert_F$ (sup is over the three matrices)

My idea is blend SVD decomposition and the fact that the Frobenius norm of a matrix is the squared singular values and hence LHS would be cropped as only $r$ singular values would contribute to the sum. But I can't make it formal. Do you have any hints on how to proceed?

——— Edit post @whuber comment!

OK. This was question was actually not posed correctly at all sorry. What I am looking is not really to simply show that $ \sup \Vert\mathbf{WA} \Vert_F \leq \sup \Vert\mathbf{WB}\Vert_F$ but rather to find the separating factor based on the ranked (A) vs. full ranked (B) constraint. So basically looking a result like;

$$ \sup \Vert\mathbf{WA} \Vert_F \leq \textrm{some function of} (r, \kappa) \\ \sup \Vert\mathbf{WB} \Vert_F \leq \textrm{some function of} (m, \kappa) $$

Happy to get any pointers! Thanks again :)

Answer 1

Consider the case where $p=n$ so $WA$ and $WB$ are square. Let $WB$ equal the identity matrix, and $WA$ equal the identity matrix with the $[n,n]^{th}$ entry equal to zero. The Frobenius norm of $WB$ is obviously equal to $\sqrt{n}$, and that of $WA$ to $\sqrt{n-1}$.

Now, let the $[n,n]^{th}$ entry of $WB$, label it $\sigma^2_n$, $\to 0$. Clearly $\Vert WB\Vert _F = \sqrt{n-1+\sigma^2_n} \to \sqrt{n-1}$.

We can easily extend this to the case of $r < n-1$ and nonsquare matrices; the key is the singular values, not the shape. Thus, for every $WA$, there is a $WB$ with Frobenius norm arbitrarily close to that of $WA$.

This indicates that, as long as we know nothing but the rank of the two matrices $WA$ and $WB$, we cannot show that the separating factor is greater than zero.