Norms of Primitive Idempotent

Question

I'm working on a proof on Algebraic Graph Theory. I'm almost done, except that I'm not quite sure on this step. Is this equation true? $ \Vert E_j \hat{x} - u \cdot E_j \hat{y} \Vert ^2 = \Vert E_j \hat{x} \Vert ^2- u^2 \Vert E_j \hat{y} \Vert ^2 $ .

Where Ej is a primitive idempotent.

If the equation above is true? Tips on how to show it?

Answer 1

This is false. Set $y=x$ and $u=-1$, when the LHS is positive (when $E_jx\ne0$) and the RHS is zero.