Let $A$ and $B$ be two fixed similar matrices, and $X$ and $Y$ be matrices such that $A=XBX^{-1}$ and $A=YBY^{-1}$. Clearly we cannot conclude that $X=Y$ since for instance $X$ could be a scalar multiple of $Y$. Indeed, we know that if $X$ and $Y$ are any two matrices such that $Y^{-1}X$ commutes with $B$, then $$ YBY^{-1} = YB(Y^{-1}X)X^{-1} = Y(Y^{-1}X)BX^{-1} =XBX^{-1}.$$
Question: Is this the worst that can happen? That strikes me as unlikely— I see no obvious reason that the set of "similarity matrices for $(A,B)$" couldn't be even larger. Assuming this is true, is there a nice characterization of this set? (If not, have its properties been studied by folks before?)
Motivation: I have a module homomorphism $T:\Bbb Z^n\to \Bbb Z^n$ which is naturally expressed in a specialized basis $(f_i)$, and I also computed its action with respect to the standard basis $(e_i)$. I extended it to a linear map $\Bbb Q^n\to\Bbb Q^n$ and computed its rational canonical form $R$.
In the abstracted notation above, the $A$ that I care about is $[T]_{(f_i)}$ and the $B$ that I care about is $R$. Using a particular (naive) algorithm, I can compute a particular choice of conjugating matrix to get $A$ to be in rational canonical form; that is the $X$ that I care about. The $Y$ that I care about is what this algorithm spits out if I first do a change of basis to $[T]_{(e_i)}$.
In doing so, I observed a phenomenon that, because I was wrapped up in some problem-specific details, I found very counterintuitive at the time: there is a prime number $p$ which shows up in the denominators of the entries in $Y$ but does not appear in any denominators of the entries in $X$. Considering the objection from the first paragraph, I am less mystified. But this got me to wonder if anything useful can be recovered from this general situation.
