Eigenvalues of a rotation matrix

[Lucy Yeats]

Homework Statement

Find the eigenvalues and normalized eigenvectors of the rotation matrix
cosθ -sinθ
sinθ cosθ

Homework Equations

The Attempt at a Solution

c is short for cosθ, s is short for sinθ
I tried to solve the characteristic polynomial (c-λ)(c-λ)+s^2=0, and got λ=cosθ±sinθ.
I then tried to find the eigenvectors, but only got (0, 0) and couldn't find any others. Where did I go wrong?

 

[I like Serena]

What do you get if you write down R-λI?

 

[Lucy Yeats]

I think you get:

cosθ-λ -sinθ
sinθ cosθ-λ

 

[I like Serena]

Oh, I think I see your problem.
Your eigenvalues should be λ=cosθ ± i sinθ.

And I meant for you to write down R-λI with λ=cosθ + i sinθ...

 

[Lucy Yeats]

I'm not sure where the i comes from.

I did this to get the eigenvalues:
(c-λ)(c-λ)+s^2=0
c^2-2λc+λ^2+s^2=0
as C^2+s^2=1,
λ^2-2cλ+1=0
λ=\frac{2c±√(4c^2-4)}{2}
=c±√(c^2-1)=c±s=cosθ±sinθ

 

[I like Serena]

You have √(c^2-1) = √(cos²θ-1)
Since cosθ≤1, this means that you're taking the square root of a negative number.
Hence the "i".I was assuming you are supposed to solve this with imaginary numbers.
If you are supposed to only find real eigenvalues, then there are none.

 

[Lucy Yeats]

Ah, I can't believed I missed such an obvious mistake! Thanks for pointing it out.

 

[I like Serena]

Okay... so you found your eigenvectors?

 

[Lucy Yeats]

Yes, are they (ia, a) and (-ia, a) for all a?
If so, how do you normalize imaginary eigenvectors?

 

[I like Serena]

Good!

The length of a vector (a,b) in $\mathbb{C}^2$ is given by $\sqrt{|a|^2 + |b|^2}$.
Normalization works the same as with real vectors.

 

[Lucy Yeats]

So you normalize a vector by writing √(x^2+y^2)=1?
But doesn't (ia)^2+a^2=a^2-a^2=0?

 

[I like Serena]

Not quite, |ia| = |a|.
The absolute value of a complex number x+iy is given by |x+iy|=√(x^2+y^2).
For the length of a complex vector, you need to use these absolute values.

 

[Lucy Yeats]

So are they (i√0.5, √0.5) and (-i√0.5, √0.5)?

 

[I like Serena]

Yep!

 

[Lucy Yeats]

Thank-you! :-)