Find the eigenvalues and eigenvectors for the matrix

[tomeatworld]

Homework Statement

Find the eigenvalues and eigenvectors for the matrix [{13,5},{2,4}]

Homework Equations

None

The Attempt at a Solution

Well eigenvalues is easy, and turn out to be 14 and 3.
So using eigenvalue 3, the two equations 10x₁ + 5x₂=0 and 2x₁ + x₂=0. Using these, I assumed the eigenvector to be [1,-2] but after putting it into Wolfram (or mathematica) it gives out the vector should be [-1,2]. Is there a difference? How should you discern between the two if not and where have I gone wrong if there is?

 

[Mark44]

Homework Statement

Find the eigenvalues and eigenvectors for the matrix [{13,5},{2,4}]

Homework Equations

None

The Attempt at a Solution

Well eigenvalues is easy, and turn out to be 14 and 3.
So using eigenvalue 3, the two equations 10x₁ + 5x₂=0 and 2x₁ + x₂=0. Using these, I assumed the eigenvector to be [1,-2] but after putting it into Wolfram (or mathematica) it gives out the vector should be [-1,2]. Is there a difference? How should you discern between the two if not and where have I gone wrong if there is?

Assuming your work is correct, there is no difference between <1, -2> and <-1, 2> as far as being eigenvectors. Each of these vectors is the -1 multiple of the other, so they are both in the same eigenspace, a subspace of dimension 1 (a line) in R².

 

[tomeatworld]

Great. So do programs like mathematica choose them at random or is there a reason it chose <-1,2> over <1,-2>?

 

[Mark44]

I have no idea.

 

[tomeatworld]

Ok, thanks for the help!