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Math Help - complex eigenvalues

  1. #1
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    complex eigenvalues

    i've been given a problem to find the eigenvalues of R = RyRz given theta = pi/2. Ry and Rz are your standard rotation matrixes, so
    the real one is simple enough, but when i try to sub in lander i get

    (0-lander)0 1

    1 (0-lander)0

    0 1 (0-lander)


    which is evaluating as -

    (0 - lander)^3 + 1 = 0

    so the real solution is 1, as it should be for a rotation. but i cant see where the complex solution could be. should i have kept the equation as trig functions maybe ?
    thanks
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  2. #2
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    The eigenvalue equation is \lambda^3-1=0. One solution is \lambda=1, so you can divide by the factor \lambda-1, giving (\lambda-1)(\lambda^2+\lambda+1)=0. The other two eigenvalues come from solving \lambda^2+\lambda+1=0. You can do this by the usual process for solving a quadratic equation, and the solutions are \lambda=-\tfrac12\pm\tfrac{\sqrt3}2i.

    Linguistic note: the Greek letter \lambda is lambda, not lander.
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