Matrices Multiplication $AB + BC = D$

Question

Suppose I have the following equation $AB + BC=D$, where $A$, $C$, and $D$ are known, how can I derive the solution for $B$?

Answer 1

This is a Sylvester equation. Suppose that $B$ has shape $m \times n$. In terms of vectorization and the Kronecker product, we have $$ (I_n \otimes A + C^T \otimes I_m)\operatorname{vec}(B) = \operatorname{vec}(D). $$ If the coefficient matrix is invertible, this can be solved as usual with $$ \operatorname{vec}(B) = (I_n \otimes A + C^T \otimes I_m)^{-1}\operatorname{vec}(D). $$