# Eigenvectors and Eigenspace

## Homework Statement

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

## Homework Equations

(A - lambda I ) v = 0

## 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
Homework Helper
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]

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
Homework Helper
(-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?

ehild
Homework Helper
You do not need to normalize in principle.

ehild

ok thanks.