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Math Help - Conjugation of matrices.

  1. #1
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    Conjugation of matrices.

    def.: SO(3) is the collection of lineair orthogonal transformations of \mathbb{R}^3, the elements are 3\times 3 matrices.

    I want to show that two element of SO(3) are conjugated if and only if they have the same trace.

    (Hint: Maby you want to use: if A \in SO(3)\ \mbox{then}\ \exists p \in \mathbb{R}^3,\ p\neq 0\ \mbox{and}\ Ap=p )
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  2. #2
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    Okay, what is the definition of "conjugate matrices"?
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    Okay, what is the definition of "conjugate matrices"?
    C'mon, this is a tough question. At least assume the OP has some knowledge!
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    Quote Originally Posted by HallsofIvy View Post
    Okay, what is the definition of "conjugate matrices"?
    Two matrices A and B are conjugated if there exists an orthogonal transformation, such that:

    A= CBC^{-1}
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  5. #5
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    Quote Originally Posted by bram kierkels View Post
    Two matrices A and B are conjugated if there exists an orthogonal transformation, such that:

    A= CBC^{-1}

    No. Two square matrices A,B are conjugate if there exists SOME invertible matrix P s.t. A=PBP^{-1}

    One direction of your problem is almost trivial: ANY two conjugate matrices, whether orthogonal or not, have the same trace (and the same determinant, and the same eigenvalues...).
    The other direction looks highly suspicious to me, but I cannot prove it isn't true.

    Tonio
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    MHF Contributor Bruno J.'s Avatar
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    Using the hint : by an appropriate conjugation, you can put any A \in SO(3) in the form \left(\begin{matrix}1 &0 & 0 \\ 0 & a & b \\ 0&c & d \end{matrix}\right) by an appropriate base change... and in fact we can also suppose \left(\begin{matrix} a &b \\ c& d\end{matrix}\right)=\left(\begin{matrix} \cos \theta &-\sin \theta \\ \sin \theta& \cos \theta\end{matrix}\right). (What this means is essentially that a rotation of three-dimensional Euclidean space is always a rotation around an axis; the axis is fixed, and its orthogonal complement is rotated like a plane. So two rotations are conjugate only if the angles of rotation are equal up to a sign.)

    Can you take it from there?

    Another way might be to use the isomorphism SO(3) \cong SU(2)/\{\pm \mbox{id.}\}, and the fact that two Möbius transformations are conjugate if and only if they have the same multiplier. But that's more complicated.
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