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Math Help - Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1

  1. #1
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    Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1

    Hi,
    I have a question about describing geometrically the action of an arbitrary orthogonal 3x3 matrix with determinant -1. I would like to know if my proposed solutions are satisfactory, or if they lack justification. I have two alternate solutions, but have little confidence in their validity. Any help would be greatly appreciated!!

    Solution 1: The orthogonal 3 x 3 matrix with determinant 1 is an improper rotation, meaning it is a reflection combined with a proper rotation. In another sense, an improper rotation is an indirect isometry, which is an affine transformation with an orthogonal matrix with a determinant 1.

    Solution 2:
    A rotation about the origin, followed by inversion through the origin (i.e. (x,y,z)-->(-x,-y,-z) ). Note that a "left-handed object" turns into a "right handed object", so "handedness is reversed" but otherwise it is just like a rotation.

    Thanks in advance!
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  2. #2
    MHF Contributor
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    Re: Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1

    Hey abscissa.

    I think the first solution is good. Basically you can decompose your transformation into two matrices one which is det +1 (rotation) and one which is -1 (reflection matrix).

    The other reason why I think its good IMO is because it specifically talks about the orientation aspect which is the critical difference between +1 and -1 orthogonal matrices (in terms of the determinant).
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