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Basis expansion

  1. Oct 29, 2014 #1
    I know that the matrices {[itex]\Gamma^{A}[/itex]} obey the trace orthogonality relation [itex]Tr(\Gamma^{A}\Gamma_{B})=2^{m}\delta^{A}_{B}[/itex]

    In order to show that a matrix M can be expanded in the basis [itex]\Gamma^{A}[/itex] in the following way

    [tex]M=\sum_{A}m_{A}\Gamma^{A}[/tex]
    [tex]m_{A}=\frac{1}{2^{m}}Tr(M\Gamma_{A})[/tex]

    is it enough to just substitute the first equation for M in the second, and work out that the RHS is indeed equal to [itex]m_{A}[/itex] (using the orthogonality), or is this just a mere verification, and not a proof?
     
  2. jcsd
  3. Oct 29, 2014 #2

    Fredrik

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    What you're proving there is that if a vector ##v## is in the subspace spanned by an orthonormal set ##\{e_i\}##, then ##v=\sum_i \langle e_i,v\rangle e_i##.

    To prove that an arbitrary ##v## in your vector space is in the subspace spanned by that orthonormal set, you will have to do something else. You will have to write down a definition of that specific vector space, and then use it somehow.
     
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