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Matrix Proof

  1. Oct 22, 2004 #1
    How do you prove that the product of two orthogonal matrices is orthogonal? I know that a matrix can be written in component form as [tex]A=a_{jk}[/tex] and that for an orthogonal matrix, the inverse equals the transpose so [tex]a_{kj}=(a^{-1})_{jk}[/tex] and matrix multiplication can be expressed as [tex]AB=\Sigma_ka_{jk}b_{kl}[/tex]. I think that is all I need to be using, but I'm not sure where to go from there.
     
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  3. Oct 22, 2004 #2

    Galileo

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    Think of some other characteristic of orthogonal matrices.
    Think about determinants in particular.
     
  4. Oct 22, 2004 #3

    shmoe

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    As you've written it, this is incorrect. You don't take the inverse of the entries. If [tex]A=[a_{jk}][/tex] is orthogonal then [tex]A^{-1}=A^{T}=[a_{kj}][/tex].

    There's no need to go into the entries though. You can directly use the definition of an orthogonal matrix. Answer this question: what do you have to do to show (AB) is orthogonal?
     
  5. Oct 22, 2004 #4
    Well the determinant of an orthogonal matrix is +/-1, but does a determinant of +/-1 imply that the matrix is orthogonal? I know that the determinant is distributive [tex]|AB|=|A||B|[/tex], so the determinant of the product does have to be +/-1, but I don't know if that is sufficient to show that a matrix is orthogonal.
     
  6. Oct 22, 2004 #5

    shmoe

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    No, it doesn't. There are matrices with determinant +/- 1 that are not orthogonal.

    To show [tex]AB[/tex] is orthogonal, you can show directly that [tex](AB)^{-1}=(AB)^{T}[/tex]. What is [tex](AB)^{T}(AB)[/tex]?
     
  7. Oct 22, 2004 #6

    Galileo

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    You're right. I was so totally confused :redface:
     
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