When I run dot(a,b) in MATLAB, I get a very different number than $\sum a_i \times b_i$, but is that not how one takes a dot product? For instance:
A = [1+i 1-i -1+i -1-i];
B = [3-4i 6-2i 1+2i 4+3i];
dot(A,B)
% => 1.0000 - 5.0000i
A(1)*B(1)+A(2)*B(2)+A(3)*B(3)+A(4)*B(4)
% => 7.0000 -17.0000i2 Answers
$\begingroup$There is a different definition when you work with complex vectors. The dot product for complex vectors is defined as: $$\mathbf{A}\cdot\mathbf{B}=\sum_i a_i\overline{b_i}$$ Maybe this link could help: Complex dot product
$\endgroup$ 0 $\begingroup$The Wikipedia page Isai linked to basically says it all, but I think it is worth unpacking some of the definitions given there here with a bit more motivation.
To generalize the usual $\mathbb{R}^n$ dot product, what we can do is to look at the properties of that dot product, and then see if we can come up with something in $\mathbb{C}^n$ that has similar properties.
One characterization of the regular dot product is as being a "symmetric positive-definite bilinear form". Let's unpack:
symmetric: $a \cdot b$ = $b \cdot a$. This is linked to the notion of the angle between two vectors being the same regardless of order.
positive definite: $\forall a \ne 0, a \cdot a > 0$. This corresponds to our usual notion of the "size of a vector being a positive real number". Remember that a inner product like the dot product naturally induces a norm
bilinear: linear in both arguments, e.g. $(\alpha a) \cdot b$ = $\alpha (a \cdot b)$ and $(a + c) \cdot b = (a \cdot b) + (c \cdot b)$
form: in the context of linear algebra, a form is a (multi-)linear map that has codomain as the underlying field of a vector space, e.g. from $\mathbb{R}^n$ to $\mathbb{R}$, $\mathbb{C}^n$ to $\mathbb{C}$
Note however that the complex are not orderable, so this definition of "form" won't work well with "positive definiteness" for the complex numbers, unless we pick a form that maps $\mathbb{C}^n$ to $\mathbb{R}$
The standard definition of the complex product mentioned by Isai:
$$a \cdot b = \sum_i a_i\overline{b_i}$$
is one that does what it can to salvage the positive definiteness, but it has to sacrifice some of the other properties in the process:
symmetric: sacrificed, obviously, since $a_i \overline{b_i} \ne b_i \overline{a_i}$ in general.
What is maintained however is $a \cdot b = \overline{b \cdot a}$. And we also note that this version is also valid for the real forms, since for real numbers $a = \overline{a}$, so we could use it as a more general version of symmetry that is maintained in both cases.
positive definite: maintained. E.g. if we consider $\mathbb{C}^1$:
$$(a + bi) \cdot (a + bi) = (a + bi)(a - bi) = (aa + -abi + bia -bbii) = a^2 + b^2$$
So we see that this form always maps to positive real numbers, not arbitrary complex numbers, and so it makes sense to talk about positive definiteness.
bilinear: for addition is maintained:
$$a \cdot \overline{b + c} = a \cdot \overline{b} + a \cdot \overline{c}$$
but for scalar multiplication, it is maintained only if the scalars are real, otherwise:
$$a_i \overline{(\alpha + \beta i) b_i} = (\alpha - \beta i) a_i \overline{b_i} \ne (\alpha + \beta i) a_i \overline{b_i}$$
We could however cheat here once again as for symmetry, and say that the real multiplication linearity on the second argument should actually be defined as:
$$a \cdot \alpha b = \overline{\alpha} a \cdot b$$
which would work on both real and complex cases.
form: maintained obviously
This complex dot product is know as a Hermitian form.
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