Matrix Transformations around z axis

[concon]

Homework Statement

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,...}

Homework Equations

Unit vector u = x/norm(x)

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?

 

[jbunniii]

Homework Statement

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)

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?

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

 

[concon]

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

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?

 

[jbunniii]

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?

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.

 

[jbunniii]

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

 

[concon]

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

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?

 

[jbunniii]

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?

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?

 

[concon]

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?

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

 

[jbunniii]

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

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

 

[concon]

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

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?

 

[D H]

So then should u = (0,0,1) since e3 points in direction of z?

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.

 

[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:

Find a unit vector s.t. Au=u
(make sure the first nonzero is positive)

Next step is to find the other two eigenvalues:

for what λ's does Au = λu? write answer as a set {a,b,...}

 

[jbunniii]

for what λ's does Au = λu? write answer as a set {a,b,...}

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.

 

[concon]

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.

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?
So should answer just be{1}?

 

[jbunniii]

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?
So should answer just be{1}?

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