- #1
aija
- 15
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Matrix A=
2 1 2
1 2 -2
2 -2 -1
It's known that it has eigenvalues d1=-3, d2=d3=3Because 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.
2 1 2
1 2 -2
2 -2 -1
It's known that it has eigenvalues d1=-3, d2=d3=3Because 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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