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Question about Pauli Matrices

  1. Oct 17, 2015 #1
    Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices.
    I want to prove that, but I haven't been able to.
    Please, could somebody point me a book where it is proven, or tell me how to do it?

    Thanks for reading
     
  2. jcsd
  3. Oct 17, 2015 #2

    blue_leaf77

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    Just try to form arbitrary linear combination of Pauli matrices and see if the resulting matrix complies with the requirement of being called self-adjoint and has zero trace.
    $$A = c_1\sigma_1 + c_2\sigma_2 + c_3\sigma_3$$
    where the ##c##'s are real constants.
     
  4. Oct 17, 2015 #3
    Write down a general ##2 \times 2## matrix as

    [tex]\left(\begin{array}{cc} a+bi & c+di \\ e + fi & g + hi \end{array} \right)[/tex]

    Now require the matrix to be self-adjoint and traceless. What constraints does this put on ##a,b,\ldots,h##? Try to see how the resulting matrix can be written as a linear combination of the three Pauli matrices.
     
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