Let $A$ and $B$ be $n\times n$ matrices over $\mathbb{C}$ . Then $A$ and $B$ are called, as I say, $\mathbb{C}^*-similar$ if there exists a non-zero scalar $k\in\mathbb{C}^*$ and an invertible matrix $P$ such that $B=kPAP^{-1}$.
Note that it is an equivalence relation. I guess that may be defined by someone before probably. But I checked it on Google but nothing was found. So I want to ask the relevant references for this. Thanks in advance.
This just means that $A$ is similar to a non-zero multiple of $B$ (or some non-zero multiple of $A$ is similar to $B$). Now the Jordan normal form of $kB$ is obtained from the Jordan normal form of $B$ by just multiplying all the eigenvalues by $k$. (To see this, observe that $kB$ is similar to $k$ times the Jordan normal form of $B$. Then it is easy to see that all the $k$'s above the main diagonal can be changed into $1$'s without changing the similiarity type) Thus the equivalence relation you propose is that the two matrices have the same Jordan normal form up to multiplication of all eigenvalues by a non-zero factor.