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Complex dot product

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 + 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}

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

Ancestors

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