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Find the eigenvectors given the eigenvalues 
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#1
Sep2711, 02:34 PM

PF Gold
P: 836

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? 


#2
Sep2711, 02:42 PM

Mentor
P: 18,334

[tex]x_1=2x_24x_3[/tex] Try to set [itex]x_2,x_3[/itex] equal to something and see what you get. For example, set [itex](x_2,x_3)=(1,0)[/itex] and [itex](x_2,x_3)=(0,1)[/itex]. This will give rise to two linear independent eigenvectors which span the eigenspace. 


#3
Sep2711, 02:47 PM

PF Gold
P: 836

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. :) 


#5
Sep2711, 02:54 PM

PF Gold
P: 836

Thanks again.
Now, i might be pushing into some daring territory here, but might you (or someone else) be familiar with matlab? Anyway, here goes the eigenvector solution from matlab: v = 0.9701 0.9487 0.6963 0 0 0.6963 0.2425 0.3162 0.1741 Notice that the last column should correspond to: [4 0 1]^T (the ratio is what matters here) and since there are no middle zero from the matlab solution, i'm a bit uneasy that i might have made a mistake somewhere in my manual calculations, although i have doubled checked everything. 


#6
Sep2711, 05:56 PM

Mentor
P: 21,313

It looks to me like the first column, not the third column, represents a vector that is a multiple of <4, 0, 1>^T.



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