What is cross product for complex vector in $\Bbb C^n$ ? and how to prove if $u,v$ are complex: $|u\cdot v| = |u||v|$ if and only if $u= av$? do I need here to define the cross product?
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Why do you want to the cross product in $\mathbb C^n$ when there isn't really a geometric interpretation for it? Is the $*$ here supposed to be the cross product or is it the dot product? – rschwieb Sep 08 '16 at 13:36
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The cross product in $\mathbb{C}^n$... does not exist! See this question: Why is cross product only defined in 3 and 7 dimensions?
Note $\mathbb{C}^n$ is isomorphic to $\mathbb{R}^{2n}$ as vector spaces, so can never have a cross product.
GiantTortoise1729
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It seems that some properties of the cross product cannot be maintained in $\mathbb{C}^3$. But surely some of them can. For instance, treated as an algebraic formula, we can maintain most if not all of the algebraic properties: antisymmetry, distributive property, and self annihilation.
Algebraic Definition: $$\mathbf{a}\times\mathbf{b}=\langle a_2b_3-a_3b_2,\,a_3b_1-a_1b_3,\,a_1b_2-a_2b_1\rangle$$
Antisymmetry: $$\mathbf{b}\times\mathbf{a}=\langle b_2a_3-b_3a_2,\,b_3a_1-b_1a_3,\,b_1a_2-b_2a_1\rangle=-(\mathbf{a}\times\mathbf{b})$$
Distributive: $$\mathbf{a}\times(\mathbf{b}+\mathbf{c})=\langle a_2(b_3+c_3)-a_3(b_2+c_2),\,a_3(b_1+c_1)-a_1(b_3+c_3),\,a_1(b_2+c_2)-a_2(b_1+b_1)\rangle$$ Then use the distributive property of complex numbers and the definition of addition of complex vectors to write this as $(\mathbf{a}\times\mathbf{b})+(\mathbf{a}\times\mathbf{c})$
Self-Annihilation: This is from antisymmetry, $\mathbf{a}\times\mathbf{a}=-(\mathbf{a}\times\mathbf{a})$
Magnitude, taken from the usual inner product does not maintain its meaning. Specifically, unless $\mathbf{a}\in\mathbb{R}^3$, $\mathbf{a}\cdot(\mathbf{a}\times\mathbf{b})\ne0$ in general. Moreover, the property in question $|\mathbf{u}\times\mathbf{v}|=|\mathbf{u}||\mathbf{v}|$ if and only if $\mathbf{u}=a\mathbf{v}$ is clearly false because $(a\mathbf{v})\times\mathbf{v}=\mathbf{0}$.
I came across this inquiry because I was wanting to say something about the cross product of real and imaginary parts of a complex eigenvector, but to use the properties of the eigenvector, I wanted to represent and manipulate the real and imaginary parts as linear combinations of complex vectors.
AThomack
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Complex numbers have a well-defined cross product.
The complex cross product is commonly used in physics to compute torques (force $\times$ lever arm), and also the intensity of fields in electricity and magnetism (remember the right-hand rule?).
Given complex numbers a and b, where v,w,x,y are real numbers and i is the imaginary number:
$a = v + wi$ and $b = x + yi$,
the cross product $z = a \times b$ is defined as:
$z = vy - wx$
Note the disappearance of $i$ in the above formula.
The result, $z$, will be a vector along an axis orthogonal to the plane that $a$ and $b$ both lie in, and thus it will not be a member of the complex numbers. You may frequently see $z$ treated as a real number (e.g. "twelve" foot-pounds), even though it technically isn't.
Alternatively, this is the definition commonly used in physics:
$z = |a||b|sin(\theta)$
where $\theta$ is the angle between the two complex numbers.
Source: https://proofwiki.org/wiki/Definition:Vector_Cross_Product#Complex_Numbers
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I think the original question was more directed at vectors in spaces such as $\mathbb{C}^2$, $\mathbb{C}^3$, $\mathbb{C}^4$, et al. – 311411 Apr 02 '21 at 18:54