Difference equation - linear algebra

[leoj soma]

Describe how you might try to build a solution of a difference equation x_k+1=Ax_k (k = 0, 1, 2, ...) if you were given the initial x₀ and this vector did not happen to be an eigenvector.

I think I should relate x₀ to eigenvectors of A, but I'm not sure how.

 

[Ishuda]

Describe how you might try to build a solution of a difference equation x_k+1=Ax_k (k = 0, 1, 2, ...) if you were given the initial x₀ and this vector did not happen to be an eigenvector.

I think I should relate x₀ to eigenvectors of A, but I'm not sure how.

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting


I would develop the general equation for x_k first in terms of A and x₀ as possibly a hint to what must be done.

For some review you might look at
https://www.google.com/search?q=eigenvector+eigenvalue+of+matrix&ie=utf-8&oe=utf-8

For the general n by n matrix for n greater than 3, computer programs are recommended for finding eigenvalues and eigenvectors. Me, I would rather use a computer program for n=3.