calculating distance between complex vectors
G.E. Ivey
>Hi,
>I am looking for a proper way to calculate the distance between two
>vectors which consist of complex numbers. How can this be done ? The
>only idea I have so far is applying the formula of the Euclidean
>distance (or the Minkowski-metric in general) to complex numbers ?
>But I am everything but sure that this is correct ?
>Any help appreciated
Yes, that is correct. If u and v are two such vectors, the distance
between them is the length of the vector u-v.
However, note that, if u-v= <a,b,c> where a,b,c are complex numbers,
then length of u-v is sqrt(a*a^+ b*b^+ c*c^) where I am using "^" to
denote the complex conjugate.
Paul D. Hanna
> Hi,
>
> I am looking for a proper way to calculate the distance between two
> vectors which consist of complex numbers. How can this be done ? The
> only idea I have so far is applying the formula of the Euclidean
> distance (or the Minkowski-metric in general) to complex numbers ? But
> I am everything but sure that this is correct ?
>
> Any help appreciated
>
> Lars Mueller
I don't know how well-versed you are on the subject, so please do not
be insulted by the following over-simplified remarks.
If you are dealing with a pure abstraction of a 3-D vector, then it is
precisely as G.E.Ivey has put it.
However, if you are dealing with complex quaternions, as in Minkowski
4-D vectors, then the algebra is equivalent to that of spinors. Here
the 'distance' is calculated using the Minkowski metric
signature(1,-1,-1,-1).
Note that space-time intervals with imaginary distances are
'elsewhere', and can not be observed, since they are outside the
'light-cone'.
Given two events, W_1 and W_2, represented in 4-D space-time, where
W_1 = (ct_1 e_0 + x_1 e_1 + y_1 e_2 + z_1 e_3)
W_2 = (ct_2 e_0 + x_2 e_1 + y_2 e_2 + z_2 e_3)
and when these two events are defined relative to the same observer's
simultaneous space, then the length of the space-time interval between
the two events is (as you are probably aware) given by the
+square-root of:
W_1*W_2' = c^2 t_1 t_2' - x_1 x_2' - y_1 y_2' - z_1 z_2'
where the ' here indicates the conjugate transpose of the event (using
terminology borrowed from the matrix form of spinors). In the above,
I use e_0 for the identity matrix, {e_k} for the respective spinor
matrices, where
(e_k)^2 = +1 for k=0,1,2,3;
(e_0)(e_k) = +(e_k)(e_0) = +(e_k) (k=0,1,2,3)
(e_1)(e_2) = -(e_2)(e_1) = +i(e_3) (i^2 = -1)
(e_2)(e_3) = -(e_3)(e_2) = +i(e_1)
(e_3)(e_1) = -(e_1)(e_3) = +i(e_2)
so these hypercomplex units differ from Hamilton's quaternion units.
The above is a general form, but since the events are defined relative
to the same observer, the components will always be real-valued, which
means that the magnitudes of each of the components have already been
taken. So, the conjugate transpose does not change the observable:
W_2' = W_2.
This gets really deep fast, and is cause for much debate, but there is
a sense in which the event vector (4-D spinor) is only complex-valued
in some underlying reality(?); that is, an observed 4-D vector is
never complex, only real.
In fact, the Lorentz transformation of some event X (4-D real), due to
a relative velocity vector V (4-D real), may be defined by the spinor
product:
X(v) = V X V'
where the conjugate transpose of V (V') is multiplied on the right,
and so the transformed 4-D vector X(v) is always a real-valued
observable. Even though the product rules involved introduce
complex-valued components, any imaginary values are cancelled out
during the product by the conjugate transposed V'.
I realize that the above may not make any sense, since I am briefly
only touching on the subject, and not doing the topic justice.
For more information, see:
http://kmr.nada.kth.se/papers/KMR-presentations/SIGGRAPH-aug2001/GeometricAlgebra/2.Alyn.pdf
http://math.ucr.edu/home/baez/Octonions/node11.html
http://kaluza.physik.uni-konstanz.de/2MS/vn/vn/node5.html
For a real deep study, see David Hestenes' Space-Time Algebra:
http://modelingnts.la.asu.edu/html/STC.html
http://modelingnts.la.asu.edu/pdf/Proper_mechanics.pdf