Let $V$ be a finite dimensional real vector space. If $S,T∈L(V,V)$ prove that $ST$ and $TS$ have the same eigenvalues whenever $T$ is an isomorphism

Question

I know the result is true even when T is not an isomorphism but how would I show it if it was one, however.

Answer 1

We have

$$\lambda I-TS=T(\lambda I-ST)T^{-1}.$$

Can you take it from here ?