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Orthogonal Complement

  1. Nov 2, 2008 #1
    1. The problem statement, all variables and given/known data

    Consider the vector space [tex]\Re[/tex]nxn over [tex]\Re[/tex], let S denote the subspace of symmetric matrices, and R denote the subspace of skew-symmetric matrices. For matrices X,Y[tex]\in[/tex][tex]\Re[/tex]nxn define their inner product by <X,Y>=Tr(XTY). Show that, with respect to this inner product,
    R=S[tex]\bot[/tex]

    2. Relevant equations

    Definition of inner product
    Definition of orthogonal compliment
    Definition of symmetric matrix
    Definition of skew symmetric matrix

    3. The attempt at a solution
    If i can show that
    R-S[tex]\bot[/tex]=0
    will it be sufficient and how do i go about it?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 2, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    What do you mean by [itex]R- S^{\bot}= 0[/itex]? To show that [itex]R= S^{\bot}[/itex] you must show that the inner product of any member of R with any member of S is 0, that's all.
     
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