Solving for matrices AB=CA

Question

We have square matrices A, B, C such that = and both main diagnols of B and C are all 0's. Is there any way of solve for A given B and C?

I am trying to avoid using the Kronecker product matrix equation identity.

Answer 1

By the theory of the Sylvester equation, there is a nonzero solution if and only if $B$ and $C$ have a common eigenvalue. If $\lambda$ is such an eigenvalue, with eigenvectors $u$ and $v$ for $B^T$ and $C$ respectively, then $A = v u^T$ satisfies $AB - CA = \lambda v u^T - \lambda v u^T = 0$.