= Complex Vector =

A '''complex vector''' is a vector whose members are [[Calculus/ComplexNumbers|complex numbers]].

<<TableOfContents>>

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== Description ==

A complex vector is simply a vector of [[Calculus/ComplexNumbers|complex numbers]].

{{attachment:complex.svg}}



=== Properties ===

The double [[Calculus/ComplexNumbers#Complex_Conjugates|conjugate]] of a vector is the original vector: ''a̿ = a''.

Conjugation is distributive.
 * {{attachment:dist1.svg}}
 * {{attachment:dist2.svg}}



=== Distance ===

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.

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''.



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CategoryRicottone