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Continuity argument?

  1. Sep 28, 2005 #1
    What's a continuity argument? For example, a question asks to prove that the determinant of a rotation matrix is always 1 using a continuity argument?
  2. jcsd
  3. Sep 28, 2005 #2
    Anyone? How would i prove in general that the det of a rotation matrix is 1?
  4. Sep 28, 2005 #3

    Tom Mattson

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    I am unsure of what you mean by a continuity argument as it pertains to matrices and determinants. But you can straightforwardly show that the determinant of a rotation matrix is 1 by writing down the matrix and taking its determinant.

    For instance, a clockwise rotation by an angle [itex]\theta[/itex] about the [itex]z[/itex] axis is described by the following matrix:

    [tex]R_z(\theta)=\left[\begin{array}{ccc}\cos(\theta)&sin(\theta)&0 \\ -\sin(\theta)&\cos(\theta)&0 \\ 0&0&1 \end{array}\right ][/tex]

    It's a piece of cake to show that [itex]det(R_z(\theta))=1[/itex].
    Last edited: Sep 28, 2005
  5. Sep 28, 2005 #4
    I'm not sure what the question means by continuity argument either. That's why I asked.

    I know could just take the determinant of the rotation matrix. Since by euler's rotation theorem any rotation matrix M can be expressed as rotations over three perpendicular axis, det M = det R1 * det R2 * det R3 = 1, where R1, R2, and R3, are just the general rotation matrices for rotating over x,y, and z axis. But I wasn't sure if this 'proof' is sufficiently rigorous and general.
  6. Sep 30, 2005 #5


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    When asking questions like this one it would be nice to define what you are talking about. In this case a rotation matrix. The definition
    Rotation matrix: A matrix for which det(A)=1
    would be very helpful

    Another definition is
    Rotation matrix: An isometric orientation preserving linear transform
    by isometric I mean the inner product defined by A matches the one defined by I (idenity matrix)
    where ' is the adjoint (conjugate transpose or if A is real transpose)
    immediately we have
    This only insures abs(det(A))=1
    so we turn to the orientation preserving bit
    That is fancy talk, but it means that I is our prototype rotation
    that is
    would be a "bad" or improper rotation
    we want for A a rotation and B any matrix
    which again does not teach us much
    I said we want I to be a prototype
    det(I)=1 but again we want to go further
    say h is a small number and we wish to construct an almost rotation
    we have
    thus we require
    we could also make better almost rotations
    The ultimate result being the actual rotation
    of course
    Thus rotation matricies are of the form exp(A) where A'=-A thus have det=1
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