If $\alpha\beta$ be a projection, is $\beta\alpha$ necessarily also a projection?

Question

Let $V$ be an inner product space finitely generated over $\mathbb{C}$ and let $\alpha$ and $\beta$ be self-adjoint endomorphisms of $V$ satisfying the condition that $\alpha\beta$ is a projection. Is $\beta\alpha$ necessarily also a projection?

I am stuck on this problem. Any hint would be appreciated.

Answer 1

Hint:

If $\alpha \beta$ is a projection than we have $(\alpha \beta)^2=\alpha \beta$

so,if $\alpha$ is invertible, we have: $$ (\beta \alpha)(\beta \alpha)=\alpha^{-1}\alpha(\beta \alpha)(\beta \alpha)=\alpha^{-1}(\alpha\beta) (\alpha\beta) \alpha=\alpha^{-1}\alpha\beta \alpha=\beta \alpha $$ so: $(\beta \alpha)^2=\beta \alpha$