Eigenvectors Problems with -1? I got an answer a few days ago but i dont understand what it means how can it be -1 and 1.

I have a problem finding the eigenvector for the following.

original matrix

[5 1]

[1 5]

lambda = 4 and 6

just for 4

[1 1] [x] = [0]

[1 1] [y] = [0]

1x + 1y = 0 so y=-1

1x + 1y = 0 so y=-1

i know the answer is meant to be:

[-1]

[1]

can someone explain how this answer is obtained and were i went wrong in detail.

previous answer

x + y = 0 does not imply y = -1, it implies y = -x

so your vector should be

[x]

[-x]

but we can get rid of the x (by factoring it out) giving

[1]

[-1]

you could also pull out a -1 giving

[-1]

[1]

## Answers (2)

You are right up to where you have 1x + 1y = 0.

There are always infinitely many eigenvectors for each eigenvalue. They all lie along the same line.

The line in this case is the line x + y = 0. Pick any point on this line to get an eigenvector.

For instance, let x = 1. Then y = -1. This gives the eigenvector (1, -1).

just for 4

[1 1] [x] = [0]

[1 1] [y] = [0]

x + y = 0 so y= -x

x + y = 0 so y= -x

now,

[x] = [x]

[y] = [-x]

[x] = x[1]

[y] = x[-1]

where x can take all possible values.