For square real matrices $A$ and $B$, if $A$ is nonsingular, we have: $$AB=TA$$ with $T=ABA^{-1}$. My question is about the case when $A$ is singular. What are the conditions to guarantee the existence of the matrix $T$ such that $AB=TA$. How to find the solution $T$ in terms of $A$ and $B$.
Thanks,