If $AX=BX$ where $A$, $B$, $X$ are all square matrices, when can I calim $A$ = $B$?

Question

My matrix knowledge is on the rusty end and this question has bothered me for a while now recently. I have always assumed that if $A$, $B$, $X$ are all square matrices, and when $AX=BX$ where $X$ is invertible, I can claim $A$ = $B$ by multiplying $X^{-1}$ on both sides. This is obviously false in the case of eigendecomposition of a square matrix. $AP = DP$ where $D$ is a diagonal marix with eigen values of $A$ on the diagonal and $P$ is a square matrix with its columns being the eigenvectors of $A$. In this case I obviously cannot claim $A = D$ just because $P$ is invertible. So what are the conditions where I can apply this technique?

Answer 1

It is true that if $X$ is invertible than $AX = BX$ implies $A=B$.

I think you're a bit confused in your expression for the diagonalizable matrix. It's $A P = PD$.