$A$ and $B$ are similar.why there are $X$ and $Y$ s.t $A=XY$ and $B=YX$

Question

let $A,B \in {M_n}$, $A$ and $B$ are similar.why there are $X$ and $Y$ s.t $A=XY$ and $B=YX$ . (NOTE: $X$ or $Y$ is nonsingular)

Answer 1

Two matrices are called similar if \begin{equation} B = X^{-1} A X \end{equation} For some (invertible) $n \times n $ matirx $X$. Also, we can write \begin{equation} A=Y^{-1} B Y \end{equation} From which we can see that \begin{eqnarray} A &=& Y^{-1} X^{-1} A X Y \\ &=& (XY)^{-1} A XY \end{eqnarray} Do you think you could take it from here?

Answer 2

Note that if $$ B = S^{-1}AS $$ For some invertible matrix $S$, then we have $$ A = S(S^{-1}A)\\ B = (S^{-1}A)S $$