Complex coordinate space ( $C^{n}$ )

Complex coordinate space of dimension 2 ( $C^{2}$ )

This section is about the definition of the dot product over

C^{n}

, which extends the definition of the dot product over

R^{n}

The complex dot product is defined as:

\sum a_{i} \overline{b_{i}}

E.g. in

C^{1}

(a + b i) \cdot (c + d i) = (a + b i) (\overline{c + d i}) = (a + b i) (c - d i) = (a c + b d) + (b c - a d) i

We can see therefore that this is a form, and a positive definite because:

(a + b i) \cdot (a + b i) = (a a + b b) + (b a - a b) i = a^{2} + b^{2}

Given:

x = \sum_{k = 1}^{n} a_{k} + b_{k} i \in C^{n}, a_{k}, b_{k} \in R

the norm ends up being:

∣ x ∣ = \sum_{k = 1}^{n} a_{k}^{2} + b_{k}^{2}

E.g. in

C^{2}

∣ (2 + 3 i, - 1 + 5 i) ∣ = 2^{2} + 3^{2} + (- 1)^{2} + 5^{2} = 4 + 9 + 1 + 25 = 39

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