Find $B$ if $AB=BC$ and $A,C$ are invertible Suppose $A$ and $C$ are known invertible complex matrices of possibly different orders. If $B$ is an unknown matrix of appropriate order such that $AB = BC$, then how could one solve for $B$?
We can solve the problem theoretically and then compute it for low dimensions.

Consider $\phi: M(n, \mathbb{K}) \to M(n, \mathbb{K})$ mapping $X \mapsto AXC^{-1}-X$ and find the kernel.
We can solve the problem theoretically and then compute it for low dimensions.

Consider $\phi: M(n, \mathbb{K}) \to M(n, \mathbb{K})$ mapping $X \mapsto AXC^{-1}-X$ and find the kernel.