How can I obtain $B = B^{T}$ if $A = 2 B − I$ is an isometry and $B^2 = B$?

Question

Prove that matrix $A=2B−I$ is an isometry (where $A^{2}=I$) $\iff$ $B$ is an orthogonal projector, i.e., $B^2=B=B^{T}$.

For now, I've just proved that $B=B^2$ using $A=2B−I$. But I have no idea how I can obtain $B=B^{T}$. Can you guys give me a hint?

Answer 1

Hints:

• From $B^2 = B$, it follows that $A^2 = I$.
• From the fact that $A$ is an isometry, it follows that $A^TA = I$.