Here's another matrix algebra question, sorry if I'm coming at these incorrectly, but this kind of thing really isn't my forte :(
Lets say we have the equation:
$0 = -2 \mathbf{u}^{T} \mathbf{Z} \mathbf{v} + \mathbf{v}^{T} \left( \mathbf{Z} \mathbf{\Sigma} + \mathbf{\Sigma} \mathbf{Z} \right) \mathbf{v}$
The goal is to solve this equation for the matrix $\mathbf{Z}$. Right now I just get bogged down at
$\left(2 \mathbf{u}^{T} - \mathbf{v}^{T} \mathbf{\Sigma} \right)\mathbf{Z} = \mathbf{v}^{T} \mathbf{Z} \mathbf{\Sigma}$
and I'm stuck because you can't exactly take the inverse of a vector to move it to the other side.
The only piece other information to go on is that both $\mathbf{Z}$ and $\mathbf{\Sigma}$ are symmetric matrices, however they are not commutative, $\mathbf{Z \Sigma} \neq \mathbf{\Sigma Z}$
What is the correct analytical approach to this problem? Is there one?
Thanks!