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## Find the eigenvectors given the eigenvalues

The problem statement, all variables and given/known data
This is the matrix A, which i need to find the eigenvalues and eigenvectors.
3x3 matrix
5 6 12
0 2 0
-1 -2 -2

The attempt at a solution
I have found the eigenvalues to be: 1, 2, 2.
So, the final eigenvalues are : 1 and 2.

Now, i found the eigenvector for eigenvalue = 1, which is:
3x1 column matrix:
[-3 0 1]^T

But for the eigenvalue = 2, i am stuck, as these are the system equations that i have before me:

3x1 + 6x2 + 12x3 = 0
-x1 - 2x2 - 4x3 = 0

I made x1 the subject of formula: -2x2 - 4x3
And then i'm not sure how to proceed. But i'm going out on a limb here, so please correct me.

Let x2 = 1 and x3 = 0

Then i get this 3x1 column matrix:
x2[-2 1 0]^T

Let x3 = 1 and x2 = 0
I get another 3x1 column matrix:
x3[-4 0 1]^T

So, all the eigenvectors in a 3x3 matrix P, are:
-3 -2 -4
0 1 0
1 0 1

Is this correct?? Most importantly, is my method correct? Is there a better method?
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 Quote by sharks The problem statement, all variables and given/known data This is the matrix A, which i need to find the eigenvalues and eigenvectors. 3x3 matrix 5 6 12 0 2 0 -1 -2 -2 The attempt at a solution I have found the eigenvalues to be: 1, 2, 2. So, the final eigenvalues are : 1 and 2. Now, i found the eigenvector for eigenvalue = 1, which is: 3x1 column matrix: [-3 0 1]^T But for the eigenvalue = 2, i am stuck, as these are the system equations that i have before me: 3x1 + 6x2 + 12x3 = 0 -x1 - 2x2 - 4x3 = 0 I made x1 the subject of formula: -2x2 - 4x3 And then i'm not sure how to proceed. But i'm going out on a limb here, so please correct me. Let x2 = 1 and x3 = 0 Then i get this 3x1 column matrix: x2[-2 1 0]^T Let x3 = 1 and x2 = 0 I get another 3x1 column matrix: x3[-4 0 1]^T Is this correct??
Your equations reduce to one equation:
$$x_1=-2x_2-4x_3$$

Try to set $x_2,x_3$ equal to something and see what you get. For example, set $(x_2,x_3)=(1,0)$ and $(x_2,x_3)=(0,1)$. This will give rise to two linear independent eigenvectors which span the eigenspace.
 Recognitions: Gold Member Thanks for your help, micromass. Could you please check on my final solution which i edited at the end of my first post above. BTW, you really have the best degree in the world. :)

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