## Eigenvectors and Eigenspace

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,
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Recognitions:
Homework Help
 Quote by jjones1573 2 v1 + 16 v2 = 0 -v1 - 8v2 = 0 so v1 = 8v2
You made a sign error. v1=-8v2

ehild
 Oh yeah thats right thanks. Is it as simple as the vector is: [-8v2] [v2] and the eigenspace is: span [-8] [1]

## Eigenvectors and Eigenspace

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.
 Recognitions: Homework Help (-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
 Oh thanks. Do I need to normalise the vectors or is it fine to just find the two vectors and give that?
 Recognitions: Homework Help You do not need to normalize in principle. ehild
 ok thanks.

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