complex eigenvalues

• November 17th 2008, 09:30 AM
def77
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
• November 17th 2008, 11:15 AM
Opalg
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.