Inverse Power Method and Eigenvectors

tatianaiistb

1. The problem statement, all variables and given/known data

The Markov matrix A = [.9 .3; .1 .7] has eigenvalues 1 and .6, and the power method u_k=A^ku₀ 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 .6^k)?

2. Relevant equations



3. 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?!
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

tatianaiistb

Also, I found that the eigenvector corresponding to the eigenvalue 1 is [3 1]^T. Still confused though... Not sure how to proceed.

tatianaiistb

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!