So, I have this algebra problem:
I have an equation of the form $$M_{ij}=\sum_{A,B}C_{ij}^{AB} X^{AB} \equiv \mathrm{Tr}\big[C_{ij}X\big]$$ where upper upper-case letters are some kind of indices and lower lower-case ones are some others. $M_{ij}$ and $C_{ij}^{AB}$ are known matrices of different types I have to solve this equation in order to find $X_{ij}$.
Well, despite it may looks like a simple linear system, it is not. I don't know how to invert it and find the $X$ matrix. Reasonably, I think it could always be solved if the dimensionality of $M$ equals that of $X$, that is, if I have enough equations for the independent variables of my problem. Is it true?
Can someone give me some tips or suggestions on how to work out this problem?
Thank you all in advance!
EDIT: probably, the $X$ matrix inside the trace should be transposed:$$M_{ij}=\mathrm{Tr}[C_{ij}X^T],$$but still.
The problem can be easily transformed to a linear system via vectorize the matrix. Make use of isomorphism $\mathrm{vec}:\mathbb{R}^m\otimes\mathbb{R}^n\cong\mathbb{R}^{m\times n}$, we can write $m=\mathrm{vec}~M, x=\mathrm{vec}~X$ as column vectors and $c$ as a matrix corresponding to $C$. Or $$m=cx$$ After obtaining the solution $x$ for the above equation, you can obtain $X$ by rearrangement of $x$.