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Compute the angle between (1,0,i,i,i) and (i,i,i,0,i) in C5

We define the angle \theta between two vectors A and B in \mathbb{C}^n by the formula

    \[ \theta = \arccos \frac{\frac{1}{2} (A \cdot B+ \overline{A \cdot B})}{\lVert A \rVert \lVert B \rVert}. \]

Compute the angle between A = (1,0,i,i,i) and B = (i,i,i,0,i) in \mathbb{C}^5.


We compute as follows:

    \begin{align*}  \theta &= \arccos \frac{\frac{1}{2} ( A \cdot B + \overline{A \cdot B})}{\lVert A \rVert \lVert B \rVert} \\[9pt]  &= \arccos \left( \frac{\frac{1}{2}(2-i + (2 + i))}{4} \right) \\[9pt]  &= \arccos \left( \frac{1}{2} \right) \\[9pt]  &= \frac{\pi}{3}. \end{align*}

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