how to determine eigenvectors of 2x2 matrices like MatLab does?

Hi I am attempting to determine the eigenvalues and eigenvectors of a 2x2 matrix.

So how to do the following Matlab instruction "manually":

>> A = [1 1; 1 2]

A =

1 1
1 2

>> [V, D] = eig(A)

V =

-0.8507 0.5257
0.5257 0.8507


D =

0.3820 0
0 2.6180

>>

I already managed to determine the diagonal matrix D with the eigenvalues but
how can I get the matrix v that contains the eigenvectors?

solve for X for each eig. value: (A-(eig. val.)I)X=0

so by inserting 0.38 into the equation I get:
0.62x1 + x2 = 0
x1 + 1.62x2 = 0
sorry for the stupid question but
while trying to solve this I cannot get one of the already determined eigenvectors using MatLab?

try to work with the exact eigenvalues (3+-sqrt(5))/2

no the problem is:
it's all zero
0.62x1 + x2 = 0 (I)
x1 + 1.62x2 = 0 (II)

by extending (I) with 0.62^-1 I get
x1 + 1.62x2 = 0
so (II) - (I) is zero at all !!! so how to
get

-0.8507 0.5257
0.5257 0.8507 ??

Of course both equations are linear dependent (because the determinant is zero after subracting the eigenvalues).

Just solve

0.62x1 + x2 = 0

hmmh sorry but how to get such a vector out of:

0.62x1 + x2 = 0 it's an equation with 2 unkown components x1 and x2
where is the trick I don't take into account until now?
So what brings me from 0.62x1 + x2 = 0
to V = -0.8507 0.5257 0.5257 0.8508 ?

let us say x1=k, then x2=-0.62k
eigenvevtor=[k -0.62k] which is equivalent to [-0.8507 0.5257]

ok i already realized there is a connection between these values
but why did MatLab chose -0.8507 0.5257 it seems to be normalized to 1 sqrt(x1^2 +x2^2)?
why can't I just say hej k= is -1 and so
the vector would be [-1 0.62]????

every multiple of [-0.8507 0.5257] is an eigenvector.

so why does MatLab indicate this vector as my eigenvector and no multiple of it?

Matlab has to make a choice, don't know why they choose this vector.

thx I think that's enough I have to know (for now) !