I can's seem to understand the last part of it.
If I want to calculate the eigen values to:
|6 -2|
|-2 3|

|6 -2| - λ|1 0|
|-2 3| |0 1|

|6-λ -2|
|-2 3-λ|

(6-λ)(3-λ)-(-2)(-2) = 0
λ1 = 7, λ2 = 2

λ=7:
|-1 -2|
|2 -4|

λ=2:
|4 -2|
|-2 1|

And that will be:
| 7 2 |
|(-2,1) (1,2)|

What I don't understand is how I get the second row to be (-2,1) and (1,2)

:/

I'm not familiar with that form of including vectors as matrix elements, but those appear to be the eigenvectors corresponding to each eigenvalue. If you already know how to find the eigenvectors, you should be all set. If not, read on.


For a 2x2 matrix with two unique eigenvalues, A-λI should always be of the form:

[ a  b]
[ka kb]

where k is a nonzero constant. The corresponding eigenvector can then be expressed as any multiple of (-b a). So for λ=7, (-b a) is (2 -1), which is equivalent to the eigenvector (-2 1), but multiplied by -1. And for λ=2, (-b a) is (2 4), which is equivalent to the eigenvector (1 2) multiplied by 2; the latter is generally preferred as it is in lower/simpler terms.


If you need to be able to compute eigenvectors for larger matrices, this technique for 2x2's won't cut it and you'll need to use the concepts of kernel and reduced row echelon form from linear algebra. If this is something you need help with as well, let me know and I can continue with the full method.