How to solve matrix equation $AXB+CX=D$ for $X$? If it is not solvable, are there any numerical methods to do it?
2026-04-11 19:30:27.1775935827
Solve matrix equation $AXB+CX=D$
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You can apply the vec operation to both sides of the equation to obtain a linear system $$\eqalign{ (B^T\otimes A + I\otimes C)\,{\rm vec}(X) &= {\rm vec}(D) \cr }$$ whose solution is $$\eqalign{ {\rm vec}(X) &= (B^T\otimes A + I\otimes C)^{-1} \,{\rm vec}(D) \cr }$$ Or, if the inverse does not exist, you may need to resort to the pseudoinverse.