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Math Help - Orthogonal 3x3 matrix with determinant -1

  1. #1
    Senior Member Pinkk's Avatar
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    Orthogonal 3x3 matrix with determinant -1

    What does such a matrix do geometrically? I'm having a hard time visualizing this; I used some basic examples but I am not seeing a general pattern. Any help would be appreciated.

    Edit: From the examples I've seen it looks kinda like a glide reflection, but again, I'm having a hard time coming up with the general case.
    Last edited by Pinkk; March 22nd 2011 at 09:17 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    What does such a matrix do geometrically? I'm having a hard time visualizing this; I used some basic examples but I am not seeing a general pattern. Any help would be appreciated.
    Orthogonal matrices act as isometries...since you took a course in geometry I know you you know what this means. Now, what do you think the negative determinant corresponds to?
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  3. #3
    Senior Member Pinkk's Avatar
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    Well, it preserves length, so it's definitely a rigid motion. It obviously can't be a rotation since all rotations have determinant 1. A determinant of -1 means it reverses orientation, so it's gonna be a reflection, or a rotation and a reflection. I think it's the latter but I really don't know how to go about finding it and proving it.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    As a consequence of a well known theorem, if A\in\mathbb{R}^{3\times 3} is orthogonal and \det A=-1 then, A is orthogonally similar to a matrix of the form:

    B=\begin{bmatrix}{-1}&{0}&{\;\;\;0}\\{\;\;0}&{\cos \alpha}&{-\sin \alpha}\\{\;\;0}&{\sin \alpha}&{\;\;\;\cos \alpha}\end{bmatrix}\quad (\alpha\in\mathbb{R})

    Besides,

    B=\begin{bmatrix}{-1}&{0}&{0}\\{\;\;0}&{1}&{0}\\{\;\;0}&{0}&{1}\end{b  matrix}\begin{bmatrix}{1}&{0}&{\;\;\;0}\\{0}&{\cos \alpha}&{-\sin \alpha}\\{0}&{\sin \alpha}&{\;\;\;\cos \alpha}\end{bmatrix}

    Now, you can conclude.
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