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

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• Dec 2nd 2011, 06:59 AM
rollinator
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?
• Dec 2nd 2011, 07:02 AM
wnvl
Re: how to determine eigenvectors of 2x2 matrices like MatLab does?
solve for X for each eig. value: (A-(eig. val.)I)X=0
• Dec 2nd 2011, 07:10 AM
rollinator
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?
• Dec 2nd 2011, 08:03 AM
wnvl
Re: how to determine eigenvectors of 2x2 matrices like MatLab does?
try to work with the exact eigenvalues (3+-sqrt(5))/2
• Dec 2nd 2011, 08:13 AM
rollinator
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 ??
• Dec 2nd 2011, 08:35 AM
wnvl
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
• Dec 2nd 2011, 08:40 AM
rollinator
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 ?
• Dec 2nd 2011, 08:44 AM
wnvl
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]
• Dec 2nd 2011, 09:18 AM
rollinator
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]????
• Dec 2nd 2011, 10:14 AM
wnvl
Re: how to determine eigenvectors of 2x2 matrices like MatLab does?
every multiple of [-0.8507 0.5257] is an eigenvector.
• Dec 2nd 2011, 11:24 AM
rollinator
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?
• Dec 2nd 2011, 11:49 AM
wnvl
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.
• Dec 2nd 2011, 12:02 PM
rollinator
Re: how to determine eigenvectors of 2x2 matrices like MatLab does?
thx I think that's enough I have to know (for now) !