Matrix A=(adsbygoogle = window.adsbygoogle || []).push({});

2 1 2

1 2 -2

2 -2 -1

It's known that it has eigenvalues d1=-3, d2=d3=3

Because it has 3 eigenvalues, it should have 3 linearly independent eigenvectors, right?

I tried to solve it on paper and got only 1 linearly independent vector from d1=-3 and 1 from d2=d3=3.

The method I used was:

[A-dI]v=0

and from this equation I used Gaussian elimination to find v1, v2 and v3

Even wolfram alpha finds only 1 solution from this:

http://www.wolframalpha.com/input/?i=-x+++y+++2z+=+0,+x+-+5y+-+2z+=+0,+2x+-+2y+-+4z+=+0

^

this is the system of equations from [A-3I]v=0(3 is the eigenvalue d2=d3)

I don't see any way to get 2 linearly independent vectors from this solution

y=0, z=x/2

all i get is vectors

t*[2 0 1]T, t is a member of ℝ

here's matrix A in wolfram alpha: http://www.wolframalpha.com/input/?i={{2,+1,+2},+{1,+2,+-2},+{2,+-2,+-1}}

It shows that there is an eigenvector v3 = [1 1 0]T, but i don't see how to get it. Obviously my way to solve this problem doesn't work, so what did I forget to do in my solution or what did I do wrong and why doesn't it work this way?

PS. I'm not sure if this should be in the homework section, because this is more like a general problem and I don't understand why doesn't it work the way i tried to solve it. Matrix A could be any matrix with two equal eigenvalues.

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# Finding eigenvectors of a matrix that has 2 equal eigenvalues

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