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Thread: What's the significance of the eigenvalues+eigenvectors of rotation transformation?

  1. #1
    s3a
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    What's the significance of the eigenvalues+eigenvectors of rotation transformation?

    Here is the question to which I am referring, along with its solution.:
    Question #4:
    https://www.docdroid.net/CtqTsdX/e2.pdf

    Question #4's Solution:
    https://www.docdroid.net/CtqTsdX/e2.pdf#page=3

    The solution seems to try to get the reader to think about the significance of the computations sought, but it doesn't seem to specify what that significance is, and I'd like to know, so I asked someone, and I was told that the eigenvalue 1 is a generator of the rotation axis, since RX = 1X = X means that the point X is not moved by the rotation R and that the other two eigenvalues don't have such an obvious representation, and that that just generate the plane in which the rotation happens.

    Unfortunately, I'm still not clear on the matter.

    I do understand how the point X is not moved by rotation R, but I don't get how that is a "generator of the rotation axis" (other than the fact that rotating in a plane perpendicular to a certain axis is a rotation about that axis). Also, about the other two eigenvalues, I understand that there generally are rotations happening in a plane (such as rotating about the z axis being equivalent to rotating on the xy plane), but I don't see how the eigenvalues generate planes (in which rotations happen).

    Could someone please help further clarify this in my mind?

    Any input would be GREATLY appreciated!
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  2. #2
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    Re: What's the significance of the eigenvalues+eigenvectors of rotation transformatio

    Hey s3a.

    If a matrix is diagonalizable then what you do with the eigenvectors is align a space so that it can be scaled in each coordinate system [by unentangling everything] and then rotating it back to the original space.

    Every matrix does what is called a basis transformation on a vector and if you use eigenvectors then you are doing a rotation to get something in an R^n like form [remember that in R^n every single variable is independent and has no relationships with the other variables].

    Think about taking a system that has linear relationships where you describe every variable as a linear combination of other variables and try and find a way to make all of those variables independent with a coordinate transformation and that will give you an idea of what that diagonalization decomposition is doing.
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