inequality on matrix Hadamard Products $\|A \odot X\|_F$

Question

I have two matrix with same size $A$ and $X$, $A$ is a binary matrix. $X$ is nonnegative matrix. Is there any inequality show that $\|A \odot X\|_F <= f(A) \|X\|_F$, how to find $f(A)$?

my goal is pull A out of frobenius norm.

Thanks

Answer 1

Since $A$ is binary, when it is zero, a term in the sum vanish, when it is one, it term remains.

\begin{align}\|A \odot X\|_F^2 &= \sum_{i=1}^n \sum_{j=1}^n (A_{ij}X_{ij})^2 \le \sum_{i=1}^n \sum_{j=1}^n (X_{ij})^2 = \|X\|_F^2\end{align}

Hence, we can set $f(A)=1$.