I have some matrices $A \in \mathbb{R}^{m \times 1}$, $B \in \mathbb{R}^{m \times p}$, and $C \in \mathbb{R}^{n \times m}$, where $n < m$.
$A$ represents a physical system of $m$ variables where the quantity $A' B$ is conserved.
I would like to reduce the dimensionality of $A$ from $m \times 1$ to $n \times 1$ by performing a linear transformation on it involving the encoder matrix $C$, without violating the conservation property above.
In other words, I'd like to find function $f(C, X)$ such that $A' B = f(C, A)' f(C,B)$.
I initially thought that it would be $f(C, X) = \mathrm{softmax}(C) X$, but it turns out that doesn't work.
Any help would be greatly appreciated!