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Conjugate of a matrix and of a function

  1. Aug 24, 2009 #1
    Hello,

    Working without complex numbers a conjugate of any function in a LVS is always the same thing. A conjugate of any matrix in a LVS is very often not the same thing though. I am just confused as to why functional spaces rely on complex numbers for the conjugate to have any importance and a matrix does not.
     
  2. jcsd
  3. Aug 25, 2009 #2

    HallsofIvy

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    Where did you see that? The "conjugate" of a matrix is just the matrix with the entries replaced by there complex conjugates. If M is a matrix with all real entries then the conjugate of M is just M itself.

    You may be confusing "conjugate" with the "conjugate transpose" or "Hermitian transpose" of a matrix: swap rows and columns and take the conjugate of each entry. Of course, if M has all real entries, it "conjugate transpose" is just its transpose.
     
    Last edited: Aug 26, 2009
  4. Aug 27, 2009 #3
    So guess my question is if functions are a different represenation of a matrix why is there no option to transpose a function?
     
  5. Aug 28, 2009 #4

    HallsofIvy

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    In what sense is a function a "different representation of a matrix"? Are you talking about representing linear functions represented by a matrix?
     
    Last edited: Aug 28, 2009
  6. Aug 28, 2009 #5
    In my QM class Operator functions are said to be like a matrix.
     
  7. Aug 28, 2009 #6

    Landau

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    I don't fully understand your question, but maybe you'd like to hear about the adjoint of a linear transformation.

    Let [tex]V,W[/tex] be inner-product spaces, let [tex]T\in L(V,W)[/tex] be a linear transformation, and [tex]T^*\in L(W,V)[/tex] its adjoint. This means that [tex]\langle Tv,w \rangle=\langle v,T^*w \rangle[/tex] for all [tex]v\in V,w\in W[/tex]. Then, the matrix of [tex]T^*[/tex] with respect to orthonormal bases of [tex]V[/tex] and [tex]W[/tex] is just the conjugate transpose of the matrix of [tex]T[/tex] with respect to these bases. As mentioned earlier, the conjugate transpose of a matrix is just the transpose (interchange rows and colums) of the matrix with all entries replaced by their complex conjugates.
     
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