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Easy method to show orthogonality of a matrix

  1. Nov 29, 2011 #1
    Hi, I know how the properties of an orthogonal matrix, the transpose ot the matrix is equal to its inverse. The problem is that the teacher gaves me a 3x3 matrix expressed in terms of many cosines and sines of three angles, I want to know how can I prove that the matrix is orthogonal without having to do the product between the matrix and its transpose.

    Thanks for your help.
  2. jcsd
  3. Nov 29, 2011 #2
    You can always check that the columns form an orthonormal basis. This implies that your matrix is orthogonal. So you only need to calculate a few inner products.
  4. Nov 29, 2011 #3

    Ben Niehoff

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    But this is actually the same as multiplying the matrix with its transpose. In fact it's the exact same sequence of operations.

    OP, I don't think there are any shortcuts to this problem unless you know some fancy group theory. Just multiply it out. All the sines and cosines should either cancel or combine into [itex]\sin^2 a + \cos^2 a = 1[/itex].
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