Proof of generalised eigenvectors

rickywaldron

Hi this is my first time posting on here so hopefully I get it right.

Given the linear system x'(t) = Ax(t)' with an eigenvalue (lambda) of algebraic multiplicity 2 and geometric multiplicity 1 (repeated root), one solution is w.exp(lambda t) and the other w.t.exp(lambda t) + u.exp(lambda t). Show that u satisfies (A-lambda.I)u = w.

We are given this equation that we have to show u satisfies in the note, but I can't work out how to show it....I'm not even sure if the question requires a full proof but here is my attempt.

Started with (A-lambda.I)u = w
Expand.....A.u - lambda.u = w
Multiply through by e(lambda.t).....A.u.e(lambda.t) - lambda.u.e(lambda.t) = w.e(lambda.t)

Now if u.e(lambda.t) was a solution to the original equation id be able to solve by subbing in to original equation and integrating, and this would work. So I feel like I'm close but can't quite workout where to go next?

Thanks