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Here's my problem, what is the motivation or idea of getting these generalized eigenvectors such that they come from computing the null spaces of (A-tI) where t is the eigenvalue with greater algebraic multiplicity than geometric and then taking the powers of the null spaces of the above matrix (a-tI) until its stabilizes at j such that dim(ker(A-tI))^j=dim(ker(A-tI))^x where x>j, that is until the dimension of the null space becomes constant after taking a finite number of powers.

I've realized that if

**(A-tI)v=0 then what we are doing is something like**

*(1)***(A-tI)w=v, where w will be the first generalized eignevector because when we sub back into**

*(2)***we get (A-tI)^2w=0 and thus if we take ker(A-tI)^2 we will see what w is in this space and w will be the eigenvector of (A-tI)^2 with eigenvalue t and at some point this will stabilize.**

*(1)*My problem is that I don't know why we do this, why

**generalized eigvectors work? Essentially, in most textbooks I've seen that they just put it down that we do this and when we have J=B^-1AB it means that B will consist of the generalized eigenvectors of A but thus far I cannot see why this works.**

__will__I would be very grateful if anyone could help. Thanks.