Choosing Free Variable for Generalized Eigenvector

[rugerts]

Homework Statement

Find general solution of DE

Relevant Equations

Eigenvector and eigenvalue eqns

As you can see from my eigenvalues, here I've got a repeated roots problem. I'm wondering if it matters which variable I can choose to be the free variable when I'm solving for the generalized eigenvector. I think both are equally valid but they look different from one another and I'd like to know the reason behind why either choice would be fine.
Thanks for your time

 

[fresh_42]

I think you have a typo in your last matrix at $(1,1)$ and thus wrong eigenvectors.

Anyway, we have a two dimensional eigenspace to the eigenvalue $-1$, so the eigenvectors span the entire vector space. How you set the parameter doesn't matter, as long as you keep the linear independent.

 

[pasmith]

If the eigenspace is the entire space, there's no reason not to use the standard basis.

 

[fresh_42]

If the eigenspace is the entire space, there's no reason not to use the standard basis.

You are right and i was wrong. We have only a one dimensional eigenspace, spanned by a single vector.
The eigenspace is annihilated by $(A+1)$ whereas the other basis vector of $\mathbb{R}^2$ is only annihilated by $(A+1)^2$.

$\operatorname{ker}(A+1)= \operatorname{span}(1,\frac{1}{2})$ and $\operatorname{ker}(A+1)^2 = \mathbb{R}^2$