Question

Let A = -3 -8 23/3
2 5 -13/3
0 0 2/3

Let w = (3,0,1)
Find the limit of A^(k)w⃗ as k approaches infinity.

Answer 1

I found the eigenvalues and eigenvectors of the matrix but i don't know hot to apply these to figure out the limit

Answer 2

I don't know the answer so take everything I say with a grain of salt. I think if you apply a transformation repeatedly on a vector it converges on the largest eigenvector.

lets say the eigenvectors are u1, u2, u3 so for some constants w=c1*u1+c2*u2+c3*u3

now if you apply A every term in this decomposition is multiplied by its own eigenvalue. Actually, I think this is the main idea.

Answer 3

thank you! completely forgot about dominant eigenvalues