# Choosing Free Variable for Generalized Eigenvector

#### rugerts

Problem 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.

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#### fresh_42

Mentor
2018 Award
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

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

#### fresh_42

Mentor
2018 Award
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 $\operatorname{ker}(A+1)^2 = \mathbb{R}^2$

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"Choosing Free Variable for Generalized Eigenvector"

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