I have an equation of the form $\vec{x}^{T} \ M \ \vec{y}$. I need something of the form $N \ \vec{x}$, where $N$ will be some function of $\vec{y}$ and $M$. The vector has been moved over to the other side.
Is there a straightforward way to do this in general?
I can't make a ton of assumptions about $M$ (it's square and I might be able to make it hermitian, but it probably won't be real or symmertic).
This is only possible because $\vec x^TM\vec y = a$ is a scalar (or more properly a $1{\times}1$ matrix) and so $a^T = a$. Thus $$ \vec x^TM\vec y = (\vec y^TM^T\vec x)^T = \vec y^TM^T\vec x = (M\vec y)^T\vec x. $$