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Orthogonal matrices form a group

  1. Nov 6, 2015 #1
    1. The problem statement, all variables and given/known data

    Show that the set of all ##n \times n## orthogonal matrices forms a group.

    2. Relevant equations

    3. The attempt at a solution

    For two orthogonal matrices ##O_{1}## and ##O_{2}##, ##x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x = x^{T}O_{2}^{T}O_{2}x = x^{T}x = x^{2}.##

    So, closure is obeyed.

    Matrix multiplication is associative.

    The identity element is the identity matrix.

    ##x'^{2} = (O^{-1}x)^{T}(O^{-1}x) = x^{T}(O^{-1})^{T}O^{-1}x = x^{T}(O^{T})^{-1}O^{-1}x = x^{T}(OO^{T})^{-1}x = x^{T}x = x^{2}##.

    So, the inverse of any orthogonal matrix is an orthogonal matrix.

    Is my answer correct?
     
  2. jcsd
  3. Nov 6, 2015 #2

    andrewkirk

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    Looks good to me
     
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