Help in Finding Eigenvectors Associated with Complet Eigenvalue

In summary, the conversation discusses the attempts to find the Eigenvector using both Matlab and by hand. The individual has successfully found the Eigenvector associated with a real Eigenvalue, but is struggling to find the Eigenvector for complex Eigenvalues. They have tried using Gauss Elimination, but are unsure of what norm to use to get the same answer as Matlab. They are seeking clarification and assistance with their approach.
  • #1
tehipwn
16
0
The last matrix at the bottom of the second page is the Eigenvector found using Matlab.

I'm trying to find it by hand. I found the Real Eigenvector associated with L=76.2348. But I've tried to find the Eigenvector's for the complex Eigenvalues for a while and can't get the answer given by Matlab. I might be doing the row operations and solving for x1,x2,x3 correctly but then using the wrong norm to get the Matlab answer?

Any help would be much appreciated.

The Attempt at a Solution



The attempt is in the attachments.

Note:
On the second page, the equation:

(-3.5476+j316.915)*x2 - 83.33*x3 = 0

should have been:
(-3.5476+j316.915)*x2 + 83.33*x3 = 0

But I tried working it from that and it still didn't work out. So I must be doing something fundamentally wrong.
 

Attachments

  • advanced_control_hw_eigenvectors.pdf
    514.1 KB · Views: 310
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  • #2


My question only pertains to the second page of the pdf. The first page consists of simply finding the Eigenvalues, and then the Eigenvector for the real Eigenvalue.

To refine my question, is the method of performing Gauss Elimination the correct method for finding the complex Eigenvector?

If so, what norm should be used to get the Eigenvector given by Matlab as shown in the final matrix of the second page of the pdf?

I hope this clarifies things.

Thank you.
 

Related to Help in Finding Eigenvectors Associated with Complet Eigenvalue

1. How do I find the eigenvalues and eigenvectors of a matrix?

To find the eigenvalues and eigenvectors of a matrix, you can use the method of diagonalization. This involves finding the characteristic polynomial of the matrix, solving for its roots (which will be the eigenvalues), and then using those eigenvalues to find the corresponding eigenvectors. The eigenvectors can also be found by using the eigenvalue-eigenvector equation, which is (A-λI)x=0, where A is the matrix, λ is the eigenvalue, and x is the eigenvector.

2. What is the importance of eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are important in many areas of mathematics and science, particularly in linear algebra and differential equations. They are used to understand the behavior and properties of linear transformations and systems, and they also have applications in areas such as quantum mechanics, computer graphics, and data analysis. In short, eigenvectors and eigenvalues help us to understand and analyze complex systems in a more simplified and organized way.

3. Can all matrices have eigenvectors and eigenvalues?

No, not all matrices have eigenvectors and eigenvalues. In order for a matrix to have eigenvectors and eigenvalues, it must be a square matrix (same number of rows and columns). Additionally, the matrix must be diagonalizable, meaning it can be transformed into a diagonal matrix through a similarity transformation. If a matrix does not meet these criteria, it will not have eigenvectors and eigenvalues.

4. How do I know if I have found all the eigenvectors and eigenvalues of a matrix?

To ensure you have found all the eigenvectors and eigenvalues of a matrix, you can use the eigenvalue-eigenvector equation and check if all the resulting vectors are linearly independent. If they are, then you have found all the eigenvectors. Additionally, the number of distinct eigenvalues of a matrix is equal to its order (number of rows or columns), so if you have found the same number of eigenvalues as the order of the matrix, you have found all of them.

5. Can I use software or calculators to find eigenvectors and eigenvalues?

Yes, there are many software programs and calculators that can help you find the eigenvectors and eigenvalues of a matrix. Some popular options include MATLAB, Mathematica, and Wolfram Alpha. These tools can save time and effort in finding eigenvectors and eigenvalues, especially for larger and more complex matrices.

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