This section is about the definition of the <dot product> over <\C^n>, which extends the definition of the <dot product> over <\R^n>.

Some motivation is discussed at: https://math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169

The complex dot product is defined as:
$$
\sum a_i \overline{b_i}
$$

E.g. in $\C^1$:
$$
(a + bi) \cdot (c + di) = (a + bi) (\overline{c + di}) = (a + bi) (c - di) = (ac + bd) + (bc - ad)i
$$

We can see therefore that this is a <form (mathematics)>, and a positive definite because:
$$
(a + bi) \cdot (a + bi) = (aa + bb) + (ba - ab)i = a^2 + b^2
$$

Just like the usual <dot product>, this will be a <positive definite symmetric bilinear form> by definition.


Complex dot product