Quick Eigenvectors Question

Printable View

October 17th 2011, 01:36 AM
Paymemoney

Quick Eigenvectors Question

Hi
The following question ask to find eigenvector:

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

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

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

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

not included in answer
$\left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

P.S
October 17th 2011, 01:48 AM
alexmahone

Re: Quick Eigenvectors Question

Quote:

Originally Posted by Paymemoney

Hi
The following question ask to find eigenvector:

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

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

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

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

not included in answer
$\left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

P.S

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

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

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

$A\left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]=1\left[ \begin{matrix} -1 \\ 1 \\ 0 \end{matrix} \right]$
October 17th 2011, 03:12 AM
Paymemoney

Re: Quick Eigenvectors Question

maybe my question is not clear. Is $\left [ \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right ]$ an acceptable answer?
October 17th 2011, 03:59 AM
Deveno

Re: Quick Eigenvectors Question

is (1,1,0) an eigenvector? yes, corresponding to the eigenvalue 5 (see post #2).
October 17th 2011, 08:25 AM
HallsofIvy

Re: Quick Eigenvectors Question

Quote:

Originally Posted by Paymemoney

Hi
The following question ask to find eigenvector:

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

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

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

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

not included in answer
$\left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$

P.S

I certainly don't understand your question. You say that "i don't understand why they choose these two values instead of the other one, which is the repeated eigenvalue 1". Which two "values" are you saying they give?
$\left[ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right]$
certainly is included, as an eigenvector of eigenvalue 5, not 1.