vectors, matrices, complex numbers, can they be one? Watch

Toasticide
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so i've heard of a complex vector space, a vector with complex components.

does this mean you can have a complex-vector-matrix?
if it's real, do u have a special way of denoting it?
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zetamcfc
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(Original post by Toasticide)
so i've heard of a complex vector space, a vector with complex components.

does this mean you can have a complex-vector-matrix?
if it's real, do u have a special way of denoting it?
Meaning a matrix with complex values?
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Kyx
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(Original post by Toasticide)
so i've heard of a complex vector space, a vector with complex components.

does this mean you can have a complex-vector-matrix?
if it's real, do u have a special way of denoting it?
Yes it is possible


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artful_lounger
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stress tensors come to mind...
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RichE
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(Original post by artful_lounger)
stress tensors come to mind...
stress tensors wouldn't have complex entries, surely?
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artful_lounger
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(Original post by RichE)
stress tensors wouldn't have complex entries, surely?
p sure there are some for e.g. EM that do

tbh complex numbers have a nasty tendency to crop up everywhere you don't want them to without being invited, as sadly neat derivations of such things which avoid them don't tend to actually be that common so usually they're being number crunched in FORTRAN (or MATLAB if your supervisor was educated in the last 10 years ) where they'll probably pop up but get whisked away in some step
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RichE
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(Original post by artful_lounger)
p sure there are some for e.g. EM that do
EM? electromagnetism? energy-momentum?
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artful_lounger
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(Original post by RichE)
EM? electromagnetism? energy-momentum?
EM is electromagnetism. I don't think I've ever seen it refer to energy momentum after 6th form

Magnetic materials have all kinds of awkwardness involved, or so I'm led to believe by my former flatmate who is soon to have a PhD in the area (who is incidentally trying to find as many ways as possible to include lyrics from ICPs "miracles" in his thesis so..."soon to have" may be more variable depending how many he throws in xD )
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atsruser
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(Original post by Toasticide)
so i've heard of a complex vector space, a vector with complex components.

does this mean you can have a complex-vector-matrix?
if it's real, do u have a special way of denoting it?
Yes, you can have matrices and vectors with complex components. Google "matrix over complex numbers" for more details.

As for the special way of denoting it: I think usually not. You just state somewhere that you are dealing with matrices over the complex numbers - I don't think there's any standard notation for them, however.
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atsruser
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(Original post by artful_lounger)
p sure there are some for e.g. EM that do
Which tensors are you thinking of? The EM field tensor doesn't have complex components, and that's the most important one in classical EM.

In fact, I'd be surprised if any modern treatment of EM requires complex coefficients - maybe they cropped up when people used to write x_4=ict?
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atsruser
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(Original post by artful_lounger)
EM is electromagnetism. I don't think I've ever seen it refer to energy momentum after 6th form
He's referring to the energy-momentum tensor (AKA stress-energy tensor), as used in GR.
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username2769500
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Probably can have it depending on what you want but I would assume it would be either vector or complex number because they're for computing or mapping.
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atsruser
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(Original post by RichE)
stress tensors wouldn't have complex entries, surely?
I think the only place where matrices/tensors with complex coefficients will crop up unavoidably (as opposed to being used for ease of algebra or whatever) will be in quantum mechanics e.g. as in the Pauli spin matrices and spinors.
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DFranklin
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(Original post by atsruser)
I think the only place where matrices/tensors with complex coefficients will crop up unavoidably (as opposed to being used for ease of algebra or whatever)
This may not contradict your "ease of algebra", but in many theoretical areas, working in an algebraically closed field makes life so much nicer that allowing matrices to have complex coefficeints is pretty much the norm.

In the other direction it's perhaps worth noting that by considering \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix} as an analogue to \sqrt{-1} and similar tricks you can do a lot of "complex like" stuff only using real matrices.
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atsruser
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(Original post by DFranklin)
This may not contradict your "ease of algebra", but in many theoretical areas, working in an algebraically closed field makes life so much nicer that allowing matrices to have complex coefficeints is pretty much the norm.

In the other direction it's perhaps worth noting that by considering \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix} as an analogue to \sqrt{-1} and similar tricks you can do a lot of "complex like" stuff only using real matrices.
I ought to add that I was referring to the use of complex numbers in physical problems in that post. Usually they are a convenience (e.g. in classical oscillating systems problems), but I believe that in at least some aspects of QM, they, or something isomorphic to them, are unavoidable.
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