1. ## Quick Eigenvectors Question

Hi
The following question ask to find eigenvector:

$\displaystyle \left[ \begin{matrix} 3 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 1 \end{matrix} \right]$

Now the answer i got was correct according to book, however i don't understand why they choose these two values instead of the other one, which is the repeated eigenvalue 1:

eigenvalue 5
$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

eigenvalue 1 repeated value
$\displaystyle \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right]$

$\displaystyle \left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]$

$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

P.S

2. ## Re: Quick Eigenvectors Question

Originally Posted by Paymemoney
Hi
The following question ask to find eigenvector:

$\displaystyle \left[ \begin{matrix} 3 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 1 \end{matrix} \right]$

Now the answer i got was correct according to book, however i don't understand why they choose these two values instead of the other one, which is the repeated eigenvalue 1:

eigenvalue 5
$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

eigenvalue 1 repeated value
$\displaystyle \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right]$

$\displaystyle \left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]$

$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

P.S
Let $\displaystyle A=\left[ \begin{matrix} 3 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 1 \end{matrix} \right]$.

$\displaystyle A\left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]=5\left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

$\displaystyle A\left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right]=1\left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right]$

$\displaystyle A\left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]=1\left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]$

3. ## Re: Quick Eigenvectors Question

maybe my question is not clear. Is $\displaystyle \left [ \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right ]$ an acceptable answer?

4. ## Re: Quick Eigenvectors Question

is (1,1,0) an eigenvector? yes, corresponding to the eigenvalue 5 (see post #2).

5. ## Re: Quick Eigenvectors Question

Originally Posted by Paymemoney
Hi
The following question ask to find eigenvector:

$\displaystyle \left[ \begin{matrix} 3 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 1 \end{matrix} \right]$

Now the answer i got was correct according to book, however i don't understand why they choose these two values instead of the other one, which is the repeated eigenvalue 1:

eigenvalue 5
$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

eigenvalue 1 repeated value
$\displaystyle \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right]$

$\displaystyle \left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]$

$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$
$\displaystyle \left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$