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Generalized Spin Matrices

  1. Dec 16, 2008 #1
    If I am to try and derive a set of 3x3 matrices analogous to the Pauli matrices, how would I go about doing this? I want to find the basis for all complex 3x3 matrices (analagous to the 3 Pauli matrices and the identity matrix for all complex 2x2 matrices) to expand a complicated matrix into so that I can derive a simpler expression. I've only just taken a basic linear algebra class, so I am not an EXPERT in vector spaces. Can anyone help me with this?

    This is NOT homework for me, just pure theory, so I didn't post it in the homework forum.
     
  2. jcsd
  3. Dec 16, 2008 #2

    Fredrik

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    I don't know, but the Pauli matrices are easy to find by explicitly calculating the most general form of a traceless self-adjoint ([itex]A=A^\dagger[/itex]) 2×2 matrix, so maybe you can start by doing something like that.
     
  4. Dec 16, 2008 #3

    George Jones

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    I don't know how useful this is for your purposes, but 3x3 generalizations of the Pauli matrices used in physics are the Gell-Mann matrices,

    http://en.wikipedia.org/wiki/Gell-Mann_matrices.
     
  5. Dec 16, 2008 #4
    thanks a lot! that should help me a lot, i will reincarnate this forum if i need further help
     
  6. Jan 2, 2009 #5
    I am trying right now to decompose a 3x3 matrix into a basis that can be visualized as a 3-D vector. I did this for a 2x2 matrix analogous to this problem, but the 3x3 matrix I have is quite a bit more complicated. I decomposed the 2x2 with the Pauli matrices, and then made it visualizable in 2-D space. Is this process possible with a 3x3 matrix? I have been trying to use the Gell-Mann matrices, but I am not having luck putting it into a 3-D basis. Let me know if you need more information, I just want to be able to do it myself with only a little help. Could you guys please help? Thanks!
     
  7. Jan 2, 2009 #6

    HallsofIvy

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    You would have to kill it first! Are you sure "reincarnate" is the word you meant?
     
  8. Jan 2, 2009 #7
    well, the this thread could be considered dead since it is inactive, plus wouldn't restarting under the same name be considered spamming? but this does not answer my question

    do u want me to start a new thread?
     
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