He folks, here's my problem:


Let $\mathbf{A}, \mathbf{B}, \mathbf{X}\_1, \mathbf{X}\_2, \mathbf{X}\_3, \mathbf{Y}\_1, \mathbf{Y}\_2, \mathbf{Y}\_3\in\mathbb{R}^{3\times 3}$ with determinante det()=+1. The matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{X}\_1$, $\mathbf{X}\_2$ and $\mathbf{X}\_3$ are known and the relation


$\mathbf{A}\mathbf{B}=\mathbf{X}\_1\mathbf{X}\_2\mathbf{X}\_3 \mathbf{B}=\mathbf{X}\_1 \mathbf{Y}\_1 \mathbf{X}\_2\mathbf{Y}\_2 \mathbf{X}\_3\mathbf{Y}\_3$


is valid if the matrices $\mathbf{X}\_i$ and $\mathbf{Y}\_i$ have the same structure. 


**The question is:** How to determine the matrices $\mathbf{Y}\_1$, $\mathbf{Y}\_2$ and $\mathbf{Y}\_3$ such that the matrices $\mathbf{X}\_i$ and $\mathbf{Y}\_i$ have the same structure and depend on components of $\mathbf{B}$, e.g. $\mathbf{Y}\_1(\mathbf{B})$? 


I'm grateful for any suggestion.


*edit: I have extended my original posting and tried to explain the problem in more detail.*



Decomposition of one Matrix into six matrices
If $X\_1, X\_2, X\_3$ are invertible, $Y\_1$ and $Y\_2$ can be any invertible matrices and $Y\_3 = (X\_1 Y\_1 X\_2 Y\_2 X\_3)^{-1} A B$.


