Can we say that for two orthonormal vectors $\textbf{u}$ and $\textbf{w}$ that their matched components squared sum to 1? I.e. $u_i^2 + w_i^2 = 1$.
It seems obvious if you think about a pair of 2-d vectors, if you use the right basis the two orthonormal vectors can always be written as (in the new basis) $ \tilde{\textbf{u}}=[1,0] $ and $ \tilde{\textbf{w}}=[0,1] $. But I am having trouble coming up with a proof based off their orthonormality (here just in the 2-D case)
$ u_1^2 + u_2^2 = w_1^2 + w_2^2 = 1 $
and
$ u_1 w_1 + u_2 w_2 = 0$.
Hint
If $u=(a,b)$ then $v=\pm k(-b,a)$ because they are orthogonal. Now, what does it mean to be unitary? (orthonormal.)