A is a 3 x 3 matrix with eigenvectors v1 = [1 0 0], v2 = [1 1 0], v3 = [1 1 1] (all vertical) corresponding to eigenvalues x1 = -1/3, x2 = 1/3, x3 = 1, respectively and x = [2 1 2].
Find (A^k)x. What happens as k approaches infinity?
I don't even know where to begin on this one.
Thanks for the help!
First use the eigenvalue equation (Av = lambda*v) to backsolve for the values in A. You'll also have to use the property that the trace of a matrix (sum of the diagonal values) is also equal to the sum of that matrix's eigenvalues.
Once you have A, you'll see it is an upper-triangular matrix with values <= 1....and you can see what will happen if you raise this matrix to a high power. (if you can't see it, just type it into matlab and do A^5, A^6, ... until you see what is happening)