Choosing Free Variable for Generalized Eigenvector

  • Thread starter rugerts
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Problem Statement
Find general solution of DE
Relevant Equations
Eigenvector and eigenvalue eqns
IMG-2049.JPG
IMG-2050.JPG
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

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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

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

fresh_42

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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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