1. ## Complex eigenvalues/eigenvectors

Given the rotation matrix (cos x -sin x)
sinx cosx
(supposed to by a 2x2 matrix)

I have to find the complex eigenvalues and eigenvectors when sin x not=0

Dont know if i'm going the right way but I have p(L) = det(cosx-L -sin x)
sin x cosx-L

= (cosx-L)^2 - (-sinx)^2 = (cosx)^2-2Lcosx+L^2+(sinx)^2

=L^2-2Lcosx+1

I dont know where to go from here, thanks

2. Originally Posted by jpquinn91
Given the rotation matrix (cos x -sin x)
sinx cosx
(supposed to by a 2x2 matrix)

I have to find the complex eigenvalues and eigenvectors when sin x not=0

Dont know if i'm going the right way but I have p(L) = det(cosx-L -sin x)
sin x cosx-L

= (cosx-L)^2 - (-sinx)^2 = (cosx)^2-2Lcosx+L^2+(sinx)^2

=L^2-2Lcosx+1

I dont know where to go from here, thanks

Weird choice for the polynomial's unknown...L...perhaps trying to make it similar to $\lambda$ ? Anyway, evaluate the quadratic's discriminant:

$\Delta =4\cos^2x-4=4(\cos^2x-1)=-4\sin^2x$ , so the (necessarily complex non-real...why?) roots are $\frac {2\cos x\pm 2\sin x\,i}{2}=\cos x \pm i\sin x =e^{\pm ix}$ .

Take it from here now.

Tonio