I have an matrix form as:
$$ \mathbf{A} \mathbf{X} \mathbf{B}+\mathbf{C} (\mathbf{X} \odot \mathbf{X}) \mathbf{D} $$
Where $\odot$ is the element-wise matrix product. And $\mathbf{A} \mathbf{X}$ simple denotes the matrix-matrix product.
I am trying to re-write the equation as
$$ f(\mathbf{A},\mathbf{B}, \mathbf{C}, \mathbf{D}) \mathbf{X} $$
or
$$ f(\mathbf{A},\mathbf{B}, \mathbf{C}, \mathbf{D}) \mathbf{X} g(\mathbf{A},\mathbf{B}, \mathbf{C}, \mathbf{D}) $$
Is it possible? my goal is to put $X$ together, and solve a linear system equation.
Thanks
The form $f({\bf A}, {\bf B}, {\bf C}, {\bf D}) {\bf X}$ implies that the expression is linear in ${\bf X}$.
However, ${\bf C}({\bf X}\odot {\bf X}){\bf D}$ is not linear in ${\bf X}$, since e.g. for ${\bf X},{\bf X}'=$
$$ {\bf C}\big((+) \odot (+)\big){\bf D} = 4{\bf C}{\bf D} \neq 2{\bf C}{\bf D} = {\bf C}(\odot ){\bf D} + {\bf C}(\odot ){\bf D} $$
Hence it is impossible to write $g({\bf A}, {\bf B}, {\bf C}, {\bf D}, {\bf }) = {\bf A} {\bf X} {\bf B}+{\bf C} ({\bf X} \odot {\bf X}) {\bf D}$ in the form you propose. Now that being said, it is possible to write $g$ in "polynomial" form
$$ g({\bf A}, {\bf B}, {\bf C}, {\bf D}, {\bf X}) = \big({\bf A} \otimes {\bf B}^\big)\cdot {\bf X} + \big({\bf C}\otimes {\bf D}^\big)\cdot {\bf X}^{\odot 2} $$
Where, subject to convention, ${\bf R} \otimes {\bf S} =\big({\bf R}_{ik} {\bf S}_{jl}\big)_{ij,kl}$ and "$\cdot$" is the 2-d tensor contraction
$$\begin{aligned} \big({\bf A} \otimes {\bf B}^\big) \cdot {\bf X} = \big({\bf A}_{ik} {\bf B}_{lj}\big)_{ij,kl} \cdot ({\bf X}_{kl})_{kl} = \sum_{kl} {\bf A}_{ik}{\bf B}_{lj}{\bf X}_{kl} = {\bf A} {\bf X} {\bf B} \end{aligned}$$