Using power method to calculate dominant eigenvalue and eigenvectors

In summary, the conversation is about using the power method to find the dominant eigenvalue and its corresponding eigenvectors for matrices. The person has attempted the first question correctly and is now stuck on the second question, wondering if they should use the middle or highest value from the matrix. Another person suggests trying another orthogonal vector to see if there is a larger eigenvalue.
  • #1
savva
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Homework Statement


Use the power method to calculate the dominant eigenvalue and its corresponding eigenvectors for the matrices.
The questions are attached with this thread. I have attempted both and seem to have done the first question correctly. I am attempting the second question and am stuck as after finding one eigenvector I got the same result again. Can anybody help me out with what I have to do here?

Also, when taking out a value from the matrix, should it be the middle value or the highest value in the matrix?

Homework Equations





The Attempt at a Solution

 

Attachments

  • Dominant Eigenvalue - Power Method.pdf
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  • #2
Sounds like you found an eigenvector with eigenvalue 1, congratulations :-)

But you should probably try another, orthogonal, start vector too to see if there's another larger eigenvalue.
 

Related to Using power method to calculate dominant eigenvalue and eigenvectors

1. What is the power method and how is it used to calculate dominant eigenvalue and eigenvectors?

The power method is an iterative algorithm used to find the dominant eigenvalue and corresponding eigenvector of a square matrix. It involves repeatedly multiplying the matrix by a vector and normalizing the resulting vector until it converges to the dominant eigenvector. The dominant eigenvalue is then obtained by taking the dot product of the resulting vector and the original vector.

2. When is the power method most useful for calculating dominant eigenvalues and eigenvectors?

The power method is most useful when the matrix is large and sparse, meaning it has many elements equal to zero. It is also effective when the dominant eigenvalue is significantly larger than the other eigenvalues, and when the initial guess for the eigenvector is close to the dominant eigenvector.

3. How does the power method handle matrices with complex eigenvalues?

The power method can handle matrices with complex eigenvalues by using complex arithmetic. Instead of taking the dot product of the resulting vector and the original vector, the complex conjugate of the resulting vector is taken. This allows for the calculation of the dominant eigenvalue and eigenvector for matrices with complex eigenvalues.

4. Can the power method be used to calculate multiple dominant eigenvalues and eigenvectors?

No, the power method can only calculate the dominant eigenvalue and eigenvector of a matrix. If there are multiple dominant eigenvalues, the method will only converge to one of them. To find multiple eigenvalues and eigenvectors, other methods such as the QR algorithm can be used.

5. What are the limitations of the power method?

The power method may not converge if the initial guess for the eigenvector is not close to the dominant eigenvector, or if the dominant eigenvalue is not significantly larger than the other eigenvalues. It also cannot be used for matrices with repeated eigenvalues, as it will only converge to one of them. In addition, the power method can be affected by round-off errors and may not converge in some cases.

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