0
$\begingroup$

The $3\times3$ matrix is: $$ A=\begin{bmatrix} 2 & -2 & 3\\ -2& -1 & 6\\ 1 & 2 & 0 \end{bmatrix} $$ $\lambda_1=-5, \lambda_2=3$ (repeated) I am concerned about $\lambda=3$

So a $3\times3$ matrix has a repeated eigenvalue and another single eigenvalue. Finding the eigenvectors of the repeated eigenvalue. I just have one equation:

$x+2y-3z=0$ (3 times the same equation)

If I put zero for $x,y,z$ one at each time i got $3$ different independent eigenvectors. but the dimension of eigenspace can't exceed $2$ as the algebraic order of the eigenvalue is only $2$.

What I am doing wrong? Should I just select 2 of the 3 equations I have in random?

Edit: The eigenvectors I get are: (0 3 2), (3 0 1), (-2 1 0)

$\endgroup$
  • $\begingroup$ Can you actually tell us what the matrix is? It's hard to help you otherwise... $\endgroup$ – 5xum Jun 2 '15 at 6:23
  • $\begingroup$ not sure if you are notified but i added the matrix, thanks for feedback $\endgroup$ – fidias Jun 2 '15 at 6:36
  • $\begingroup$ OK, now one more thing: what are the three eigenvectors you get? What do you mean by ""If I put zero for $x,y,z$"? I think this is where you made your mistake, but I don't know exactly what you did so I can't help you yet. $\endgroup$ – 5xum Jun 2 '15 at 6:40
  • $\begingroup$ Can you prove your claim "got 3 different independent eigenvectors"? $\endgroup$ – wdacda Jun 2 '15 at 6:42
  • $\begingroup$ I think you are right, that's where I am wrong but I am a bit confused if you could help me clarify why I am wrong I would be glad. thank you a lot $\endgroup$ – fidias Jun 2 '15 at 6:44
0
$\begingroup$

Your work is correct. Now you have: $x=3z-2y$, and this is the eigenspace (that has dimension $2$). You can choose: $y=0,z=1$ and you have the eigenvector $(3,0,1)^T$ or $y=1,z=0$ and you have the eigenvector $(-2,1,0)^T$.

$\endgroup$
  • $\begingroup$ Okay, thank you, so the dimension is 2 and I am not allowed to put x=0 $\endgroup$ – fidias Jun 2 '15 at 6:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.