Similar matrices over a complex field question

Question

Is it true that for all $ n \times n$ matrices $A$, $B$ over $\mathbb { C}$, the matrices $AB$ and $BA$ are similar?
Give a proof or counterexample.

Answer 1

No, it is not true. For example

$$A=\begin{pmatrix}1&0\\0&0\end{pmatrix}\;,\;\;\;B=\begin{pmatrix}0&1\\0&0\end{pmatrix}$$

then

$$AB=\begin{pmatrix}0&1\\0&0\end{pmatrix}\;,\;\;\text{but}\;\;BA=\begin{pmatrix}0&0\\0&0\end{pmatrix}$$

so $\;AB\,,\,\,BA\;$ cannot be similar as they have different rank. By the way, the above is true also over $\;\Bbb Q\;,\;\;\Bbb R\;$ or any other field.