How to solve the matrix equation AX-XA=BX+XB+C

Question

I know that $$AX+XB=C $$ is equal to $$(B^T⊕A)vec(X)=vec(C)$$ Now i don't know if i can use it here like: $$AX-XA-BX-XB=C$$ $$vec(AX-XA-BX-XB)=vec(X)$$ $$ vec(AX-XB)-vec(BX+XA)=vec(X)$$ $$(-B^T⊕A)vec(X)-(A^T⊕B)vec(X)$$ $$ [(-B^T⊕A)-(A^T⊕B)]vec(x)=vec(C)$$

Answer 1

If $L_A$ and $R_A$ are the operators $X \to AX$ and $X \to XA$ respectively (linear operators on the vector space of $n \times n$ matrices), your equation is $$ (L_A - R_A - L_B - R_B) X = C $$ You'll have a solution if and only if $C$ is in the range of $L_A - R_A - L_B - R_B$. In particular, this is the case for all $C$ if $L_A - R_A - L_B - R_B$ is invertible. You can represent $L_A$ and $R_A$ as the Kronecker products $A \otimes I$ and $I \otimes A$ respectively.