Reminder on how to find eigenvectors

[eherrtelle59]

Ok everybody, it's been awhile since I've taken linear algebra. I need some help dusting off the cobwebs. (I'm trying to follow this in a paper; this isn't a homework question, but I'll be glad to move it...)

Let's say I have a matrix M = \begin{bmatrix}
-σ & σ & 0 \\[0.3em]
ρ & -1 & 0 \\[0.3em]
0 & 0 & -1
\end{bmatrix} \

Ok, now I want to find the eigenvectors (I've already found the eigenvalues.

These are λ_1=-1 and λ_2,3 = -(σ+1)/2 +/- ( (σ+1)^2 -4σ(1-ρ))^.5

Now, to find the eigenvectors, I solve these equations:

(-σ-λ)e_1 +σ e_2 =0
ρ e_1 +(-1-λ) e_2 =0
(-1-λ) e_3 =0

Now, I see that the eigenvector for λ_1 is ( 0 0 1).

Now I'm stuck. How do I find the eigenvectors for the other two?

The result in the paper I'm reading says

λ_2,3 = (σ σ+λ_2 0 ) , (σ σ+λ_3 0 )

I forget how to do this...

 

[HallsofIvy]

Essentially, just the way you did the first part, just much messier algebra!

 

[eherrtelle59]

Um...yes...

So, (-σ-λ)e_1 +σ e_2 =0

Looking at this and getting (σ+λ)e_1 =σ e_2, I would think the eigenvector is (σ+λ σ) not (σ+λ σ).