Solving Orthonormal Basis of Eigenvectors for Matrix A

[asif zaidi]

Homework Statement

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).

Homework 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

 

[quantumdude]

I find the eigenvectors as 4.2361 and -0.2361

Those are the eigenvalues.

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

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

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.

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.

 

[HallsofIvy]

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?

 

[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

 

[quantumdude]

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

 

[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

 

[quantumdude]

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

 

[HallsofIvy]

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.