I know how to plot a 2D real vector. Let's say I have a vector 'a' represented in a row matrix, a = [1 1]. I can plot it in xy plane and I will be getting a line having the equation x=y.
Similarly, I know how to plot a complex number. Let's say I have a complex number z=a+ib. I can plot in real-imag plane and I will be getting a similar line having the equation x=y.
But I don't know how to plot a complex vector 'c' represented in a row matrix, c = [1+i 1+i].
Kindly guide in plotting other alternate complex vectors such as [1+i 1-i], [1 i], [i 1]
Dear Hakeem,
V = ( a+ib, c+id,e+if)= (a,c,e)+i(b,d,f):
Any complex component corresponds to a point in R^2
A vector of 2 complex components requires a representation
in R^2 X R^2 which is homeomorphic to R^4.
A vector of 3 complex components requires a representation
in R^2 X R^2 XR^2 which is homeomorphic to R^6.
The geometry of R^4 or R^6 is not attainable in general as simple as R^3 ,
but we can imagine their projections.
For physics and to tackle some problems in physics as electromagnetics
when dealing with ( sinusoidal field), the American physicists J.Willard Gibss, (1880), invented the idea of bi-vector, where the complex vector splits into two real vectors as the following: the complex vector
V = ( a+ib, c+id,e+if)= (a,c,e)+i(b,d,f)
where the real part of V=Re(V)=(a,c,e)
and the imaginary part of V = Im(V) = (b,d,f)
( following your question V = ( a+ib, c+id) (a,c)+i(b,d) )
and applying this definition, it is easily proved that :
"all multilinear identities valid for real vectors are also valid for
complex vectors"
The geometry and algebra of such representation are available in the attached paper, hope you find it useful in your research.
Best regards
PS.
The suggested presentation is not unique, we can use the tools of differential geometry to tackle such complex vectors based on the mathematical model under consideration.
Dear Hakeem,
The real vector (1,1) has initial point at the origin and terminal point at (1,1)
it has the direction of the line y =x and not x+y=1.
The real vector v = (a, b) has two real components and R^2 is enough to show the graph of v geometrically.
In case v = (a+ib, c+id), it belongs to R^4 and you need 4 dimensions to represent v. It is equivalent to say v = (a,b,c,d).
Best regards
Issam Kaddoura Could you please share online vector plotter tools so that the things can be visualized easily. To make it easy, the vector [1+i 0] is a good choice. I just want to comprehend pictorially to have better understanding.
Dear Hakeem,
V = ( a+ib, c+id,e+if)= (a,c,e)+i(b,d,f):
Any complex component corresponds to a point in R^2
A vector of 2 complex components requires a representation
in R^2 X R^2 which is homeomorphic to R^4.
A vector of 3 complex components requires a representation
in R^2 X R^2 XR^2 which is homeomorphic to R^6.
The geometry of R^4 or R^6 is not attainable in general as simple as R^3 ,
but we can imagine their projections.
For physics and to tackle some problems in physics as electromagnetics
when dealing with ( sinusoidal field), the American physicists J.Willard Gibss, (1880), invented the idea of bi-vector, where the complex vector splits into two real vectors as the following: the complex vector
V = ( a+ib, c+id,e+if)= (a,c,e)+i(b,d,f)
where the real part of V=Re(V)=(a,c,e)
and the imaginary part of V = Im(V) = (b,d,f)
( following your question V = ( a+ib, c+id) (a,c)+i(b,d) )
and applying this definition, it is easily proved that :
"all multilinear identities valid for real vectors are also valid for
complex vectors"
The geometry and algebra of such representation are available in the attached paper, hope you find it useful in your research.
Best regards
PS.
The suggested presentation is not unique, we can use the tools of differential geometry to tackle such complex vectors based on the mathematical model under consideration.
The problem is that the complex numbers is a two dimensional mapping. A vector consisting of two complex numbers is a four dimensional map. You will never get that on a flat sheet of paper. That doesn't mean we cannot try. Consider the vector (point?) (a+bi, c+di). We could map it on a graph paper as blue dot at coordinates (a, b) and a red dot at coordinates (c, d). If you want to look at it as more of a vector draw a blue line from the origin to (a,b) and a red line from the origin to (c,d). If you want a little more perspective to it add a yellow dot at (a,c) and a green dot at (b,d), etc. You may also add a orange dot at (a, d) and a pink dot at (c,b), etc. Choice of colors is arbitrary. This should give a little more perspective. Note there are six ways to take 4 things two at a time, which is why I suggest using 6 points for vector. Note, if you have a vector as a function of t, t real, then tracing the points with various colors should be useful.
For this first we required space ¢*¢where your 2D complex vector will lie,¢² is equivalent to R⁴ so any complex 2D vector is equivalent to a vector hyperplane plane in R⁴.
Otherwise we can use iteration for linear function for example f(z)=z+2 ,or z_{n+1}=z_n + 2 so it's iteration will give use direction as vector .
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