Eigenvectors and Eigenspace

jjones1573

1. The problem statement, all variables and given/known data
Im looking at finding the eigenvectors of a matrix but also a basis for the eigenspace

A = [ 6 16 ]
[ -1 -4 ]

lambda = 4
lambda = -2


2. Relevant equations
(A - lambda I ) v = 0


3. The attempt at a solution

So with the above equation I get:

for lambda = 4

[ 6 - 4 16 ] [ v1 ] = [ 0 ]
[ -1 -4 - 4 ] [ v2 ] [ 0 ]

so

2 v1 + 16 v2 = 0
-v1 - 8v2 = 0

so v1 = 8v2

and the basis for the eigenspace is span [ 8 ]
[ 1 ]

First is that right? because when I put it into an eigenvector calculator on the web it gives me
-8 instead of 8 but I cant see how I could get to that.

Second if this is the basis for the eigenspace then how can I find the eigenvectors for the eigenvalue?

thanks,

ehild

You made a sign error. v1=-8v2

ehild

jjones1573

Oh yeah thats right thanks.

Is it as simple as the vector is:

[-8v2]
[v2]

and the eigenspace is: span

[-8]
[1]

jjones1573

Sorry I realised this should have been posted in the calculus section would it be possible to have it moved?

I think what I have put above for the eigenspace is correct? But what about the eigenvector I cant seem to understand what this is.

ehild

(-8,1) multiplied by any number is an eigenvector. You need to find the other one, which belongs to the other eigenvalue lambda=2.
The two eigenvectors are the basis of the "eigenspace". You can choose the normalised vectors as basis.

ehild

jjones1573

Oh thanks. Do I need to normalise the vectors or is it fine to just find the two vectors and give that?

ehild

You do not need to normalize in principle.

ehild

jjones1573

ok thanks.