Norm and Multiplication

Question

If $A,B,C \in \mathbb{R}^{m \times n}$, $0 \leq C_{i,j} \leq 1$, $A_{i,j},B_{i,j} \geq 0$, and $||A||_F \leq ||B||_F$, then $||C \cdot A||_F \leq ||C \cdot B||_F$ where "$\cdot$" is pairwise product.

Is it true or not?

Answer 1

This statement is false. Let A, B, and C be the following.

$$ A = \begin{bmatrix} 0.03219379 & 0.63947916 \\ 0.29375581 & 0.98938686 \\ \end{bmatrix} $$

$$ B = \begin{bmatrix} 0.64079509 & 0.87661346 \\ 0.76240589 & 0.21987217 \\ \end{bmatrix} $$

$$ C = \begin{bmatrix} 0.00205744 & 0.08024417 \\ 0.14910081 & 0.33285072 \\ \end{bmatrix} $$

Then $||A||_F = 1.215$ and $||B||_F = 1.345$, so that $||A||_F \leq ||B||_F$, but $||C\cdot A||_F = 0.336$ and $||C\cdot B||_F = 0.152$, so that $||C\cdot A||_F > ||C\cdot B||_F$, where $X\cdot Y$ denotes the pairwise/element-wise multiplication.