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?

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

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

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

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?

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

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

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

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 ??

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

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

Just solve

0.62x1 + x2 = 0

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

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 ?

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

let us say x1=k, then x2=-0.62k

eigenvevtor=[k -0.62k] which is equivalent to [-0.8507 0.5257]

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

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]????

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

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

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

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

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

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

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

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