Let $x,y,w,z$ be vectors and $A$ a matrix. Let the dimension agree so that $$ x^TAy - w^TAz $$ is well defined.
I am looking for a way to factorize the elements in a form that resemble
$$ k_1^TAk_2, $$ for some $k_1$ and $k_2$. Is there any way to do that for $x\neq y\neq w\neq z$?
For example, in the case $x = w$ is easily done by
$$ x^\top A(y-z). $$
What about the general case?
If $k_1,k_2$ are required to be vectors, then I suspect that there is no such factorization.
On the other hand, we can write
$$ x^TAy - w^TAz = \\ \operatorname{tr}[x^TAy] - \operatorname{tr}[w^TAz] =\\ \operatorname{tr}[Ayx^T] - \operatorname{tr}[Azw^T] =\\ \operatorname{tr}[A(xy^T - wz^T)^T] = \\ \operatorname{vec}(A)^T \cdot \operatorname{vec}(xy^T - wz^T). $$