step in proof: Cauchy-Scharz complex inequality 
Is there a smart way to see this? I tried writing it out for $n=1$ ($z=x+iy,w=u+iv$), expecting it to be simple, but I got:
$$
(x-\operatorname{Re}(\lambda)u+\operatorname{Im}(\lambda)v)^2+(y-\operatorname{Im}(\lambda)u+\operatorname{Re}(\lambda)v)^2
$$
Halfway throughout the expansion, I feel like this can't be the way to go? Is there a simpler approach?

With the substitution $a = z_j$ and $b= \lambda \overline w_j$ the desired identity becomes
$$
 | a - b |^2 = |a|^2 + |b|^2 - 2 \operatorname{Re}(a \overline b)
$$
and that follows easily by expanding
$$
 | a - b |^2 = (a - b)\overline{(a - b)} = a \overline a + b \overline b - a \overline b - b \overline a \, .
$$
