Multiplication of matrices and requirements of eigenvectors

Question

Suppose matrix $A$ has eigenvalue 1 with corresponding eigenvector $\mathbb{x}$. If $BA$ is to have eigenvalue of 1 with the same eigenvector, what would be the requirement or condition for it?

Answer 1

We would need to have $(BA)\mathbb{x}=\lambda\mathbb{x}$. We know that matrix multiplication is associative and that $A\mathbb{x}=\mathbb{x}$, so it should be that $\lambda\mathbb{x}=(BA)\mathbb{x}=B(A\mathbb{x})=B\mathbb{x}$. We can see that by definition $\mathbb{x}$ would also have to be an eigenvector of $B$.

Running the equality in the other direction we can see that the converse is also true; that is, if $\mathbb{x}$ is an eigenvector of both $A$ and $B$ then it is an eigenvector of $AB$.