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Math Help - Quick Eigenvectors Question

  1. #1
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    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
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  2. #2
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    Re: Quick Eigenvectors Question

    Quote Originally Posted by Paymemoney View Post
    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]
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  3. #3
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    Re: Quick Eigenvectors Question

    maybe my question is not clear. Is  \left [ \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right ] an acceptable answer?
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  4. #4
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    Re: Quick Eigenvectors Question

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

    Quote Originally Posted by Paymemoney View Post
    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.
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