Finding an eigenvector

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February 1st 2013, 01:09 PM
algorithm

Finding an eigenvector

Hi,

I'm trying to find an eigenvector of a matrix. I have λ = 1, so my matrix (A - λI) is
$[-0.5253, 0.8593, -0.1906; -0.8612, -0.5018, 0.1010; 0.1817, 0.1161, -0.0236]$

And from rows 2 and 3 I get these simultaneous equations

$-0.8612t_{1}-0.5018t_{2}+0.1010t_{3}=0$

$0.1817t_{1}+0.1161t_{2}-0.0236t_{3}=0$

I eliminate to find $t_{2}=0.225t_{3}$ and $t_{1}=-0.0137t_{3}$

Thus the eigenvector is

t= $k (-0.0137, 0.225, 1)$

But the actual answer is given as (-0.0088, 0.216, 1).

Thanks for any pointers.
February 1st 2013, 02:45 PM
Deveno

Re: Finding an eigenvector

one question: are you using decimal approximations of rational numbers?
February 1st 2013, 11:41 PM
algorithm

Re: Finding an eigenvector

Hi,

They're not approximations, just measurements.
February 2nd 2013, 10:04 AM
algorithm

Re: Finding an eigenvector

Is my technique right?
February 2nd 2013, 11:55 AM
ILikeSerena

Re: Finding an eigenvector

Looks like you are having rounding errors.

When I calculate the eigenvector for the matrix you give, I'm getting different results than either of your answers.
See for instance here: {{1-0.5253, 0.8593, -0.1906},{ -0.8612, 1-0.5018, 0.1010},{ 0.1817, 0.1161, 1-0.0236}} - Wolfram|Alpha Results
The rounding errors you have are propagating more than you may like.

To answer your question: yes, your technique is right.
Note that there are more advanced methods to keep the rounding errors to a minimum.
February 2nd 2013, 12:33 PM
algorithm

Re: Finding an eigenvector

Thanks very much for confirming. I've used more accurate values and I get a more sensible answer, still not spot-on though. Important thing is that my method is right.