# Orthonormal Basis

1. Jan 13, 2008

### asif zaidi

1. The problem statement, all variables and given/known data
My problem is I am getting a different answer than what MATLAB is giving me and I cannot determine why. Plz advise.

Find an orthonormal basis of eigenvectors for matrix A= [3 2; 2 1] (using MATLAB notation- I couldn't figure out how to put in proper matrix notation).

2. Relevant equations

I find the eigenvectors as 4.2361 and -0.2361

For eigenvalue 4.2361: [3 2; 2 1] - [4.2361 0; 0 4.2361] = [-1.2361 2; 2 -3.2361]
-1.2361x1 + 2x2 = 0
x2 = 0.6180 x1
Therefore eigenvector: (1 0.6180) and normalizing it: (0.8506 0.5257)

For eigenvector -0.2361: Eigenvector: (1 -1.6180) and normalizing it: (0.5257 -0.8506)

Therefore my orthonormal basis of eigenvectors:
(0.8506 0.5257; 0.5257 -0.8506)

First Question:
Is what the question is asking - to get an orthonormal basis of eigenvectors. Is this what I am doing?

Second Question:
I think it is but when I compare my answer to MATLAB, for eigenvector 4.2361, MATLAB gives normalized eigenvectors (-0.8506 -0.5257).
I don't understand where the negative comes from.

Thanks

Asif
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 13, 2008

### Tom Mattson

Staff Emeritus
Those are the eigenvalues.

You are obtaining an orthonormal basis of the eigenspace of the matrix.

That's OK, the basis of the eigenspace is not unique. Think back to $\mathbb{R}^2$. The standard basis is $\{<1,0>,<0,1>\}$, but an equally acceptable orthonormal basis is $\{<-1,0>,<0,1>\}$. Both bases span the space, and they are both orthonormal.

3. Jan 14, 2008

### HallsofIvy

Staff Emeritus
Is this a course on Linear Algebra/Matrix Algebra or a course on how to use MATLAB?

The eigenvalues of this matrix are $2\pm \sqrt{5}$. Do you know how to find those without using MATLAB?

You seem to be consistently confusing "eigenvalues" with "eigenvectors"- you keep calling a single number an "eigenvector".

An eigenvector corresponding to eigenvalue $2+ \sqrt{5}$ is $<2, -1+\sqrt{5}>$ and an eigenvector corresponding eigenvalue to $2-\sqrt{5}$ is $<2, -1-\sqrt{5}>$. Can you find an orthonormal pair from that?

4. Jan 14, 2008

### asif zaidi

HallsofIvy:

- The course is on Linear Algebra. I use matlab to verify my results
- I know how to compute eigenvectors/eigenvalues. It was a typing error when I called an eigenvalue an eigenvector.
- As I said in original problem, I got the eigenvalues (same as you) and calculated the respective eigenvectors. I then found an orthonormal pair from it by squaring, summing and taking square root.

- The thing I am not sure from T Mattson reply is that he is saying I am getting an orthonormal basis of eigenspace and not of eigenvector and I don't see the difference.

If you can clarify I would appreciate.

Thanks

Asif

5. Jan 14, 2008

### Tom Mattson

Staff Emeritus
The expression "orthonormal basis of eigenvectors" makes no sense. Spaces, not vectors, have bases.

6. Jan 14, 2008

### asif zaidi

Tom-
I will ask my professor but that is what he is asking in the h/w assignment.
I quote: "Find an orthonormal basis of eigenvectors" for matrix which I gave above.

Thanks

Asif

7. Jan 14, 2008

### Tom Mattson

Staff Emeritus
I'm pretty sure he meant "eigenspace", which is the space of all the eigenvectors of the matrix.

8. Jan 14, 2008

### HallsofIvy

Staff Emeritus
An othonormal basis for the vector space consisting of eigenvectors of this matrix is what you really want to say.

In this case, the 'eigenspace" is the entire space.