What are the rules for complex-component vectors and why?

Answered question

2022-01-17

What are the rules for complex-component vectors and why?

Answer & Explanation

nick1337

nick1337

Expert2022-01-19Added 777 answers

Step 1
Given complex vectors
v=(v1,,vn)
and
w=(w1,,wn),
their scalar (=dot) product is given by
v×w=j=1nvjw¯j=v1w¯1++vnw¯n
Why? Well, you want to generalize the usual dot product on R, but you also want that
v×v0
for all v, and the vector (i, 0,, 0) shows that you can't do without the conjugation. You might ask why, I don't care if
v×v0
for all v. Most people do: the expression
||v||=v×v
should define a norm and ||vw|| a metric on Cn, and taking square-roots of non-positve numbers (or even complex numbers) simply isn't well-defined. Note that
||v||=v×v=j=1n|vj|2
Having settled this, let v be a non-zero vector. Its orthogonal complement
U=v={uCn:u×v=0}
is the set of all vectors orthogonal to v. Since U is determined by the single linear equation
u1v¯1++unv¯n=0,
it is an (n1) -dimensional subspace of Cn Finding solutions is easily achieved using Gauss elimination, this will give you vectors
u1, , un1
which you can make into an orthonormal basis of U using Gram-Schmidt (note that the notation
<<u, v>>=u×v
is to be understood. The fact that you're working with PK\mathbb{C}\) and not with R is immaterial, just be careful to note that
v×(λw)=λ¯(v×w)
i.e., the dot product is conjugate-linear in the second variable.
Finally, in order to solve the equation
w×v=d
simply take any
uU=v
and put
w=u+dv×vv,
and note that
w×v=(u+dv×vv)×v=(u×v)+dv×vv×v=0+d=d

Vasquez

Vasquez

Expert2022-01-19Added 669 answers

To take the length of a complex vector you need the squared magnitudes of the components. So the vector (1+i, 1i) has length |1+i|2+|1i|2=2+2=2 It is still true that the dot product of orthogonal vectors is zero in a complex space. So once you find a B parallel to A with A×B=d you can add any vector orthogonal to A to it and the dot product will not change. But when you take the dot product, remember to take the complex conjugate of the components of B.

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