How Can I Find Eigenvalues and Normalized Eigenvectors for a Matrix?

[debjit625]

Homework Statement

Find the eigen values and normalized eigen vectors for the matrix

cosθ sinθ
-sinθ cosθ

2. The attempt at a solution
Well I did the eigen values hope they are correct but can't solve for eigen vectors

Eigen values are
λ = cosθ ± isinθ

on solving for eigen vector for eigen value λ = cosθ + isinθ ,I got
x + iy = 0 ,hence only solution x = y = 0 which is not the solution I guess (eigen vector can't be null vector)
or could I take y = i and x = 1 then it is solvable but again can I take imaginary numbers ?

Thanks

 

[Orodruin]

x + iy = 0 ,hence only solution x = y = 0

x and y are not necessarily real.

 

[debjit625]

That means I take y = i and x = 1.
Thanks.