If two complex vectors are orthonormal, does conjugating one of them preserve orthonormality?

Question

Suppose $x_1$ and $x_2$ are two orthonormal vectors with complex elements. I wonder if $x_1$ and $\bar x_2$ still orthonormal to each other? Thanks for any help!

Answer 1

The question is whether for complex numbers $x_1,x_2,\cdots,x_n$ and $y_1,\cdots,y_n$, we have the implication $$ \sum_{i=1}^n x_i \overline{y_i} = 0 \quad \Longrightarrow \quad \sum_{i=1}^n x_i y_i = 0. $$ Of course if $n=1$ this is true since $x_1 \overline{y_1} = 0$ implies that $x_1$ or $y_1$ is zero, hence conjugating doesn't change the sum. But with $n=2$ there are already counter-examples : pick $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 1 \\ i \end{bmatrix}$ and $\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} 1 \\ -i \end{bmatrix}$. Then $$ x_1 \overline{y_1} + x_2 \overline{y_2} = 1 \cdot 1 + i \cdot \left( \overline{-i} \right) = 1+i^2 = 0, \quad x_1 y_1 + x_2 y_2 = 1 \cdot 1 + i \cdot (-i) = 1-i^2 = 2. $$ Hope that helps,