Matrix Proof

  • Thread starter Hypnotoad
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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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  • #2
Galileo
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Think of some other characteristic of orthogonal matrices.
Think about determinants in particular.
 
  • #3
shmoe
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Hypnotoad said:
the inverse equals the transpose so [tex]a_{kj}=(a^{-1})_{jk}[/tex]

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?
 
  • #4
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Galileo said:
Think of some other characteristic of orthogonal matrices.
Think about determinants in particular.

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.
 
  • #5
shmoe
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Hypnotoad said:
Well the determinant of an orthogonal matrix is +/-1, but does a determinant of +/-1 imply that the matrix is orthogonal?

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]?
 
  • #6
Galileo
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shmoe said:
No, it doesn't. There are matrices with determinant +/- 1 that are not orthogonal.
You're right. I was so totally confused :redface:
 

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