Solving Orthonormal Basis of Eigenvectors for Matrix A

In summary: What is the dimension of the entire space? How does the column space of the matrix relate to its eigenspace?In summary, the conversation is about finding an orthonormal basis of eigenvectors for a given matrix. The person is using MATLAB to verify their results and has obtained the eigenvalues and eigenvectors. However, there is confusion about the terminology and the difference between an eigenspace and eigenvector. It is clarified that an orthonormal basis for the eigenspace is required, and the eigenspace in this case is the entire space. The dimension of the entire space and its relation to the column space of the matrix are also discussed.
  • #1
asif zaidi
56
0

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
 
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  • #2
asif zaidi said:
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 [itex]\mathbb{R}^2[/itex]. The standard basis is [itex]\{<1,0>,<0,1>\}[/itex], but an equally acceptable orthonormal basis is [itex]\{<-1,0>,<0,1>\}[/itex]. Both bases span the space, and they are both orthonormal.
 
  • #3
Is this a course on Linear Algebra/Matrix Algebra or a course on how to use MATLAB?

The eigenvalues of this matrix are [itex]2\pm \sqrt{5}[/itex]. 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 [itex]2+ \sqrt{5}[/itex] is [itex]<2, -1+\sqrt{5}>[/itex] and an eigenvector corresponding eigenvalue to [itex]2-\sqrt{5}[/itex] is [itex]<2, -1-\sqrt{5}>[/itex]. Can you find an orthonormal pair from that?
 
  • #4
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
The expression "orthonormal basis of eigenvectors" makes no sense. Spaces, not vectors, have bases.
 
  • #6
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
I'm pretty sure he meant "eigenspace", which is the space of all the eigenvectors of the matrix.
 
  • #8
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.
 

1. What is an orthonormal basis of eigenvectors?

An orthonormal basis of eigenvectors is a set of vectors that are both orthogonal and normalized, meaning they are perpendicular to each other and have a magnitude of 1. These vectors are associated with a square matrix and are used to simplify calculations involving the matrix.

2. Why is it important to solve for an orthonormal basis of eigenvectors?

Solving for an orthonormal basis of eigenvectors allows us to diagonalize a square matrix, which means we can simplify the matrix and make it easier to perform calculations with it. This is important in many applications, such as solving systems of linear equations or finding the eigenvalues and eigenvectors of a matrix.

3. How do you find the orthonormal basis of eigenvectors for a matrix?

The process of finding the orthonormal basis of eigenvectors for a matrix involves first finding the eigenvalues of the matrix. Then, for each eigenvalue, we solve for the corresponding eigenvector. Finally, we normalize the eigenvectors by dividing them by their magnitude to ensure they are unit vectors. The resulting set of eigenvectors will form the orthonormal basis for the matrix.

4. Can the orthonormal basis of eigenvectors be used to solve any type of matrix?

No, the orthonormal basis of eigenvectors can only be used to solve square matrices. It is also important to note that not all square matrices have a complete set of eigenvectors, so this method may not always be applicable.

5. Are there any applications of the orthonormal basis of eigenvectors in real life?

Yes, the orthonormal basis of eigenvectors has many applications in fields such as physics, engineering, and computer science. It is used in tasks such as image processing, data compression, and solving systems of differential equations. Additionally, it is used in quantum mechanics to represent the states of a quantum system.

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