Homework Help: Matrix Transformations around z axis

1. Apr 29, 2014

concon

1. The problem statement, all variables and given/known data
The matrix A transforms space by rotating counterclockwise around z axis by ∏/4.

-What is A?
-Find a unit vector s.t. Au=u
(make sure the first nonzero is positive)
-for what λ's does Au = λu? write answer as a set {a,b,...}

2. Relevant equations

Unit vector u = x/norm(x)

3. The attempt at a solution

So I already found A :

[ sqrt2/2 -(sqrt2/2) 0
sqrt2/2 sqrt2/2 0
0 0 1]

which is correct.

How do I find u?

I think once I find u do I follow the process associated with finding eigenvalues
by using det(A-λI) to find λ's?

2. Apr 29, 2014

jbunniii

Consider the equation $Au = u$. This means that ____ is an eigenvalue of $A$ and ____ is a corresponding eigenvector. (Fill in the blanks.)

3. Apr 29, 2014

concon

This seems like it might be wrong, but I did the usual procedure for finding eigenvalues and got
λ = sqrt(1/2) + sqrt(2)/2
Is this right? I think I might have solved the equation wrong?

4. Apr 29, 2014

jbunniii

That's one of the eigenvalues, assuming you meant $\sqrt{1/2} + i \sqrt{1/2}$. (It's a complex number.) But there are two others.

The equation $Au = u$ tells you what one of the eigenvalues and eigenvectors must be.

Last edited: Apr 29, 2014
5. Apr 29, 2014

jbunniii

By the way, there's an easy geometric interpretation that should be helpful: what vectors are unaffected by rotation around the z axis?

6. Apr 29, 2014

concon

Just thinking, but does that mean that 1 must be one of the eigenvalues?

We haven't been over this in class so I'm just guessing, but is a zero vector unaffected by rotation?

7. Apr 29, 2014

jbunniii

Yes, a zero vector is unaffected by rotation, but a zero vector is also unaffected by any linear transformation (matrix multiplication), so it doesn't give us much information. For this reason, eigenvectors are defined to be nonzero vectors satisfying $Av = \lambda v$ for some $\lambda$.

What other vectors are unaffected by rotation around the z axis?

8. Apr 29, 2014

concon

Wouldn't the vectors e1,e2,e3= {(1,0,0),(0,1,0),(0,0,1)} be unaffected?

9. Apr 29, 2014

jbunniii

Well, $e_1$ points in the direction of the positive x axis. Is the x axis affected by rotation around the z axis?

10. Apr 29, 2014

concon

I think that yes it would have to be affected now that I think about it.So then should u = (0,0,1) since e3 points in direction of z?

11. Apr 29, 2014

D H

Staff Emeritus
Exactly.

In three dimensional space, there is always one and only one axis that isn't affected by a non-trivial (i.e., non-identity) rotation. That's the rotation axis.

12. Apr 29, 2014

jbunniii

OK, that gives you one eigenvalue (1) and corresponding eigenvector ([0, 0, 1]). And the eigenvector is already scaled so it is a unit vector, as requested in the problem statement. So that takes care of the second question, which was:
Next step is to find the other two eigenvalues:

13. Apr 29, 2014

jbunniii

Actually, that question is worded rather poorly. It could be interpreted as referring to the $u$ from the previous part, in which case of course only $\lambda = 1$ is correct and the question seems pointless.

But I interpret it to mean "for what $\lambda$'s is there a nonzero $u$ such that $Au = \lambda u$?" In other words, find all of the eigenvalues. Probably best to check with your instructor just to make sure.

14. Apr 29, 2014

concon

yes that is what the professor meant. So we already have 1 as eigenvalue. The question specificies that only real values of λ to be included. Earlier in the thread I solved for the eigenvalues and got
λ= sqrt(1/2) + isqrt(1/2) which isn't a real number right?