I've been having some trouble with conceptually understanding the idea of a generalized eigenvector. If we have a linear operator A and want to diagonalize we get it's eigenvalues and eigenvectors but if the algebraic multiplicity of one of the eigenvalues is greater than the geometric multiplicity we can't diagonalize. Now, it makes sense that we want to make A as diagonal as possible. 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 realised that if (1) (A-tI)v=0 then what we are doing is something like(2) (A-tI)w=v, where w will be the first generalized eignevector because when we sub back into (1) 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. My problem is that I don't know why we do this, why will 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. I would be very grateful if anyone could help. Thanks.