Inverse Power Method and Eigenvectors

In summary, when dealing with the Markov matrix A, the dominant eigenvalue is 1 and the smallest eigenvalue is 0.6. The eigenvectors of A-1 are the same as those of A, with the corresponding eigenvector to the eigenvalue 0.6 being [-1 1]T. To apply the inverse power method to A-1, the eigenvector to the eigenvalue 1 is [3 1]T. Further steps for applying the power method to A-1 are not provided.
  • #1
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Homework Statement



The Markov matrix A = [.9 .3; .1 .7] has eigenvalues 1 and .6, and the power method uk=Aku0 converges to [.75 .25]T. Find the eigenvectors of A-1. What does the inverse power method u-k=A-1u0 converge to (after you multiply by .6k)?

Homework Equations





The Attempt at a Solution



Eigenvalue 1 is the dominant one when using the power method on A. However, we're interested in the smallest eigenvalue when dealing with the inverse power method, in this case .6. The eigenvalues of A-1 are:
(1/.6) and 1. According to theory, the eigenvectors of A-1 are the same as those in A.

So, the corresponding eigenvector to the value .6 is [-1 1]T.

From there, I'm simply stumped. Can anyone please help?!
 
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  • #2
Also, I found that the eigenvector corresponding to the eigenvalue 1 is [3 1]T. Still confused though... Not sure how to proceed.
 
  • #3
I know that these eigenvectors correspond to the eigenvalues of A-1, and these eigenvalues are the reciprocal of those given. Does anyone know how to apply the power method to A-1? Any ideas? Thanks!
 

1. What is the Inverse Power Method?

The Inverse Power Method is an algorithm used to find the eigenvector associated with the smallest eigenvalue of a square matrix. It is an iterative method that starts with an initial guess and improves the estimate with each iteration until it converges to the desired eigenvector.

2. Why is the Inverse Power Method used?

The Inverse Power Method is used to find the eigenvector associated with the smallest eigenvalue of a matrix. This is useful in many applications, such as in physics and engineering, where the smallest eigenvalue represents the lowest natural frequency of a system.

3. How does the Inverse Power Method work?

The Inverse Power Method works by using the power method on the inverse of the original matrix. This means that in each iteration, the original matrix is inverted and multiplied by the current estimate of the eigenvector. This process is repeated until the desired eigenvalue and eigenvector are obtained.

4. What is the relationship between the Inverse Power Method and Eigenvectors?

The Inverse Power Method is a method used to find the eigenvector associated with the smallest eigenvalue of a matrix. It is an extension of the power method, which is used to find the eigenvector associated with the largest eigenvalue. Both methods rely on the same principle of repeated matrix multiplication to converge to the desired eigenvector.

5. How accurate is the Inverse Power Method in finding Eigenvectors?

The accuracy of the Inverse Power Method depends on the choice of initial guess, the number of iterations, and the condition of the original matrix. In general, it is a highly accurate method for finding eigenvectors and can be improved by using better initial guesses and performing more iterations.

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