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Why does this method of finding a rotation angle fail?

  1. Jul 30, 2012 #1
    °I'm working on Mathematical methods in the Physical Sciences by Mary L Boas on my own, and I sometimes check Cramster (cheggit) for worked solutions when I get completely stuck, since the book is sometimes light on example.

    I was given a 3x3 matrix, and I was asked to show that it is orthogonal. Then, I calculated it's determinant which was -1, meaning it was a reflexion matrix. I used that information to find the eigenvector of λ=-1, since I knew -1 was an eigenvalue. Mary L Boas explains in her book that it is then possible to find the other eigenvectors without having to find the eigenvalues. The eigenvector of λ=-1 is the new z axis. To find the new x axis, the dot product can be used to find a vector perpendicular to z. Then, the new y axis can be used to find with the cross product of the 2 other vectors. The new axes now form a right-handed orthogonal triad. So far so good.

    I now have my 3 eigenvectors, which means I have my matrix C. (C being the matrix that diagonalizes a matrix M)

    It's a well known fact that C-1MC = D. I wanted to find D, because I could compare it to rotation matrices and figure out the rotation angle. When I found D, I found that D was equal to the identity matrix, which means that the original matrix was a reflexion, and caused no rotation.

    I understand everything up to this point. Here's what I don't understand:

    On cramster, the worked solution was wrong. Intead of calculating D and comparing it to the rotation matrices to find the rotation angle, the poster used the original matrix M. By calculating M2, the poster found that M2 = I. Thus, according to the poster, applying the matrix M twice to any matrix wouldn't change that matrix. Which means each "application" of M must correspond to 360°/2 = 180°. I know that my answer is correct because it's in the back of the book.

    Why is the method on cramster wrong? Would it work for a pure rotation matrix (without a reflexion)?? Obviously, applying a reflexion twice would cancel them out, so it's easy to see that M2 = I in this case.

    Basically, I want to know if I can use that "trick" to find the rotation angle of a matrix, or if it will be prone to mistakes like the one on the worked solution. The advantage of that solution was that it requires much less work than mine.

    Sorry for the bad english, and sorry if I used some words incorrectly. English isn't my main language.
    Last edited: Jul 30, 2012
  2. jcsd
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