Let $\circ$ denote the Hadamard product, i.e. entry-wise multiplication. Suppose we have 4 vectors $\vec{x}, \vec{y}, \vec{z}$ and $ \vec{t}$ be in $\mathbb{Z}^m$ such that $$ \vec{x}\circ\vec{y}=\vec{z}\circ\vec{t} $$ Let $A\in\mathbb{Z}^{n\times m}$ be a matrix ($m>n$) that can be multiplied by $\vec{x}$ (or any other of the above mentioned vectors) on the right. Does exist any kind of non-trivial function $f$ (for example not the constant functions) such that $$ f(A\vec{x}, A\vec{y})=f(A\vec{z}, A\vec{t}) $$ or is there a way to prove that such a function $f$ cannot exist (or has to be constant)?
Thank you!
EDIT: I am also interested if it is possible to obtain such a function $f$ when we assume some kind of restrictions on the vectors $\vec{x}, \vec{y}, \vec{z}$ and $\vec{t}$.
EDIT2: It is not a problem if the function $f$ depends onthe matrix $A$ used.