# 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?

15. Apr 30, 2014

### jbunniii

Yes, if only real numbers are allowed then the only eigenvalue is 1.