complex eigenvalues

**def77** · November 17th 2008, 09:30 AM

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

**Opalg** · November 17th 2008, 11:15 AM

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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