Common factor of 2 terms ( matrices)

Question

I have this terms (AXB+X), where A,B, and X are matrices.

How can I take matrix X as a common factor from these 2 terms.

Thanks

Answer 1

You can't directly. But it is possible in a certain way if you use Kronecker product transformation of expression $\mathbf {AXB+X}$. More precisely, you will be able to factor out not $\mathbf{X}$ but $\operatorname{vec}(\mathbf {X})$ in the following form that I borrow from the excellent Wikipedia article on the Kronecker product:

$$\operatorname {vec} (\mathbf {AXB+X} )=\displaystyle \underbrace{\left(\mathbf {B} ^{\textsf {T}}\otimes \mathbf {A} \right)}_K \,\operatorname {vec} (\mathbf {X} )+\operatorname {vec} (\mathbf {X} )$$

where $K$ is a "huge" $n^2 \times n^2$ matrix (if your matrices are $n \times n$) and $\operatorname {vec} (\mathbf {X} )$ is the $n^2 \times 1$ column vector obtained by "piling up" all the columns of $\mathbf {X} $.

For concrete examples, see this MSc Thesis.