If $\rm AX = XB$ for any $\rm X$ of constant dims $m \times n$, then does $\rm A$ exists for given $\rm B$ and how to find it?

Question

What is the name of matrix $\rm A$ with respect to matrix $\rm B$ so that $$\rm AX = XB$$ for any $\rm X$?

Does this matrix exist? How to find it?

Answer 1

Note that you can always take $X = I$, so always $A = B$; and the only matrices $A,B$ which satisfy this are $A = B = \alpha I$, where $\alpha$ is any scalar.

Answer 2

The equation is a special case of the Sylvester equation $$ AX+XB=C, $$ where we could write $-B$ instead of $B$. Here $A,B,C$ are given, and we look for solutions $X$ (as we usually do for equations). In your case, however, you only have given $B$, and you want the equation to hold for all $X$. This restricts the possibilities to find a matrix $A$ severely. If all matrices are square, then taking $X=I$ yields $A=B$, and both equal to a scalar multiple of the identity.

Edit: If it is meant that $AX=XB$ for some $X$, then you can solve the Sylvester equation to find an $X$, see wikipedia and its links.