# Homework Help: Geometric Properties from Eigenvectors

1. Oct 19, 2008

### Daniel323

1. The problem statement, all variables and given/known data
Hi! I just used MATLAB to find the eigenvalues and eigenvectors of A=[0 -1; 1 0]
I obtained the eigenvalues of 0 +/- i
and eigenvectors of v(1) = [ 0.7071; 0 - 0.7071i] and v(2) = [ 0.7071; 0 + 0.7071i]

2. Relevant equations
I'm having trouble interpreting these results in relation to the geometric properties of the linear transformation T(x; y) = [0 -1; 1 0] [x; y]

3. The attempt at a solution
As far as what the transformation does this is my explanation: It changes the sign of the y values i.e. positive to negative or negative to positive.
As far as interpreting the MATLAB results in relation to the geometric properties of the linear transformation T, I can only explain it as: the eigenvalues illustrate that the x values remain the same but the y values change in sign.

Is such an explanation correct. Is this what the transformation does.
Sorry I haven't done much mathematical explainations and not quite sure if this is how to go about it.

2. Oct 20, 2008

### tiny-tim

Hi Daniel323!

(have plus-or-minus: ± and a square-root: √ )

i] You can multiply eiegenvectors (but not eigenvalues, of course) by any factor … in this case, it's much neater if you divide by .7071 (= 1/√2 ), to give eigenvectors of v(1) = [ 1; -i] and v(2) = [1; i].

ii] If a complex number is purely imaginary, you don't have to write it as "0 + …": in this case, you can write that the eigenvalues are ±i.

iii] Hint: draw the lines through the origin along [ 1; -i] and [1; i]. What does T do to these lines? Is there a rotation? Is there a reflection?

3. Oct 20, 2008

### Daniel323

Hi tiny-tim, thanks for that.

I drew the lines on a graph and I think this is the effect (and this would be a much better explanation then my previous one);
The linear transformation T reflects the line or each point through the x axis.

Hopefully I'm right, thanks again! :)

4. Oct 20, 2008

### tiny-tim

Hi Daniel323!
Nooo … y = x, x = -y is a … ?

Hint: a reflection will have a negative determinant.

A rotation (in 2D) has no fixed line, and therefore no real eigenvalues.

5. Oct 20, 2008

### Daniel323

Ahhhh.... here's goes nothing:
The linear transformation T rotates the line or each point through the x axis?

Thanks again.

6. Oct 20, 2008

### tiny-tim

"nothing shall come of nothing …"

I'm not following you … what's a rotation through the x axis?

(this is only 2D …)

7. Oct 20, 2008

### Daniel323

Sorry about that, my mistake. I'm just horrible at mathematical explanations.

The linear transformation T causes a rotation for each vector. Is this generally a good sort of explanation for such findings?

8. Oct 20, 2008

### tiny-tim

Yes, that's fine!

Rotation reflection and translation are the correct mathematical terms for what all 2D or 3D transformations are made of.

But can I just check … a rotation of how much, and about what axis?

9. Oct 20, 2008

### Daniel323

Of course you can! I believe it is a rotation about the x-axis, but as far as how much I'm not completely sure how to say. If I was to guess I'd say a rotation of 180 degrees about the x-axis.

10. Oct 20, 2008

### tiny-tim

hmm … this is 2D, so any rotation must be about the z-axis (or a line parallel to the z-axis).

Just try a couple of test points, to see where they go.
(Actually, I should have added expansion, either generally or in one direction.)

11. Oct 20, 2008

### Daniel323

Hmm I'm kind of confused now, because it is 2D I thought we were only dealing with a x and y axis.

So now the rotation is about the z-axis, but again not sure how to describe how much of a rotation.

I tried out some points and generally understand what the transformation is doing, just can't explain it mathematically, so to speak.

12. Oct 20, 2008

### tiny-tim

ok, I agree, technically there's no z-axis, but you can rotate about the origin (or any other point), which is the same as a rotation about the z-axis.

In other words, stick a pin in the paper, to fix it to the desk, and rotate!

As for a proper mathematical explanation, any rotation about a particular axis (or point, in 2D) is defined by its angle … so how through many degrees is everything rotated?

13. Oct 20, 2008

### Daniel323

Ok so I will say it is a rotation about the origin or point by 180 degrees.
I suspect it is 180 degrees, but maybe I'm looking at it wrong and it is only 90 degrees.

14. Oct 20, 2008

### tiny-tim

Yes, it's one or the other!

Hint: stop trying to be abstract …

do a concrete example …

what happens to (1,0)?

15. Oct 21, 2008

### Daniel323

Umm I'm gonna go with a rotation about the origin by 180 degrees. Please tell me I'm finally right!

16. Oct 21, 2008

### tiny-tim

Well, 180º would send (1,0) to (-1,0), but A(1,0) = (0,1), so … ?

17. Oct 21, 2008

### HallsofIvy

No need to guess. What does that transformation change <1, 0> to?

18. Oct 26, 2008

### Daniel323

Ahh the transformation changes (1,0) to (0,1) hence the x and y value swaps at this point and therefore a rotation about the origin by 90 degrees.

I believe I am right here, thanks very much!