$S, T\in L(V)$ on $\mathbb{C}$ and $V$ is finite dimensional, both $S$ and $T$ are normal. If $ST$ is normal, prove $TS$ is normal.
Here is another post, but in this problem it didn't say $S$ and $T$ commute. If use definition, $(ST)(ST)^*=(ST)^*(ST)$, since it didn't say they commute, we cannot derive for $TS(TS)^*$ from here. Another way maybe is to show $||(TS)v||=||(TS)^*v||$, but after write them into inner product, it still get stuck due to non-commutive. Is there any hint how to start?
