For similarity of matrices, if A and B are similar, are $A = P^{-1}BP$ and $A = PBP^{-1}$ equivalent?

Question

For similarity of matrices, if A and B are similar, are $A = P^{-1}BP$ and $A = PBP^{-1}$ equivalent?

I know that matrix multiplication is not commutative, but it seems to me that it is in this specific case?

I would greatly appreciate it if people could please clarify this.

Answer 1

Well, if you set $P=Q^{-1}$, then we see

$$A=P^{-1}AP=(Q^{-1})^{-1}BQ^{-1}=QBQ^{-1}$$

So yes, they're equivalent.