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| The double [[Calculus/ComplexNumbers#Complex_Conjugate|conjugate]] of a vector is the original vector: ''a̿ = a''. | The double [[Calculus/ComplexNumbers#Complex_Conjugates|conjugate]] of a vector is the original vector: ''a̿ = a''. |
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| * If ''c = ab'', then ''c̅ = a̅b̅''. * If ''c = a + b'', then ''c̅ = a̅ + b̅''. * If ''c = a - b'', then ''c̅ = a̅ - b̅''. |
* {{attachment:dist1.svg}} * {{attachment:dist2.svg}} |
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| Note that a complex vector as ''z = [1+i 1-i]'' would, according to the conventional definition for [[Calculus/Distance|distance]], have a magnitude of 0. Only the zero vector should have a magnitude of 0 however. | Note that a complex vector as ''v = [1+i 1-i]'' would, according to the conventional definition for [[Calculus/Distance|distance]], have a magnitude of 0. Only the zero vector should have a magnitude of 0 however. |
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| Distance of complex vectors relies [[Calculus/ComplexNumbers#Complex_Conjugate|conjugation]]. In [[LinearAlgebra|linear algebra]] notation, the distance of complex vector ''z'' is equal to ''z̅^T^z''. | Distance of complex vectors relies on [[Calculus/ComplexNumbers#Complex_Conjugate|conjugation]]. Note that this is algebraically equivalent to calculating the distance of the absolute value. If ''v = a + bi = [a b]'', then the distance is ''||v|| = √(a^2^ + b^2^)''. In [[LinearAlgebra|linear algebra]] notation, ''||v|| = v̅^T^v''. |
Complex Vector
A complex vector is a vector whose members are complex numbers.
Contents
Description
A complex vector is simply a vector of complex numbers.
Properties
The double conjugate of a vector is the original vector: a̿ = a.
Conjugation is distributive.
Distance
Note that a complex vector as v = [1+i 1-i] would, according to the conventional definition for distance, have a magnitude of 0. Only the zero vector should have a magnitude of 0 however.
Distance of complex vectors relies on conjugation. Note that this is algebraically equivalent to calculating the distance of the absolute value. If v = a + bi = [a b], then the distance is ||v|| = √(a2 + b2).
In linear algebra notation, ||v|| = v̅Tv.