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Math Help - Finding eigenvalues and eigenvectors of a real matrix

  1. #1
    Senior Member chella182's Avatar
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    Finding eigenvalues and eigenvectors of a real matrix

    Okay, so the matrix is...

    \left(\begin{array}{cc}6 & 5 \\ -5 & 0\end{array}\right)

    I solved det(A-\lambda I)=0 to find \lambda (which is the method I was taught), and I ended up with \lambda=3\pm4i, but now I'm a bit stuck. I've written down...

    \left(\begin{array}{cc}6 & 5 \\ -5 & 0\end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)=(3\pm4i)\left(\begin{array}{c}  x \\ y\end{array}\right)

    ... but I can't seem to manipulate it to get the eigenvectors.
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  2. #2
    ynj
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    AX=\lambda X\Longleftrightarrow (A-\lambda I)X=0..
    you just have to solve the right side equation, which is a linear equation groups,for a specific \lambda, say \lambda=3+4i
    A-\lambda I=\left(\begin{array}{cc}3-4i& 5 \\ -5 & -3-4i\end{array}\right)
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  3. #3
    Senior Member chella182's Avatar
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    Yeah I know that... I'm not stuck on that I'm stuck on what to do once I've found the eigenvalues.
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    Quote Originally Posted by chella182 View Post
    Okay, so the matrix is...

    \left(\begin{array}{cc}6 & 5 \\ -5 & 0\end{array}\right)

    I solved det(A-\lambda I)=0 to find \lambda (which is the method I was taught), and I ended up with \lambda=3\pm4i, but now I'm a bit stuck. I've written down...

    \left(\begin{array}{cc}6 & 5 \\ -5 & 0\end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)=(3\pm4i)\left(\begin{array}{c}  x \\ y\end{array}\right)

    ... but I can't seem to manipulate it to get the eigenvectors.
    Read this: The Case of Complex Eigenvalues
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  5. #5
    Senior Member chella182's Avatar
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    Uhm, sort of thanks, but I'm still not coming up with it for my example. Exam's in an hour and a half anyway. Oh well.
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    Quote Originally Posted by chella182 View Post
    Uhm, sort of thanks, but I'm still not coming up with it for my example. Exam's in an hour and a half anyway. Oh well.
    Where exactly is your problem? For \lambda = 3 + 4i the two equations reduce to the single equation (3 - 4i) x + 5y = 0. The link I gave you tells you what to do with this.


    Ideally you would have met and resolved this sort of question earlier than a few hours before an exam.
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  7. #7
    Senior Member chella182's Avatar
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    Sorry, but I really don't think it's your place to tell me how to revise for my exams. I have been revising for this exam for MONTHS now, and the threads I posted last night were just a last-minute attempt to iron out a few creases.

    I do understand the theory of it, I just have problems reducing it to one equation, which is what I wanted explaining.

    I know you won't care, but the exam went well and this didn't matter in the end 'cause the question I was given had real eigenvalues (which I find easier to get my head around), so hooray!
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  8. #8
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    Quote Originally Posted by chella182 View Post
    Sorry, but I really don't think it's your place to tell me how to revise for my exams. Mr F says: When someone flys the woe-is-me flag with remarks like "Exam's in an hour and a half anyway. Oh well." then I think the observation I made is justified.

    I have been revising for this exam for MONTHS now, and the threads I posted last night were just a last-minute attempt to iron out a few creases. Mr F says: It's a pretty major crease.

    I do understand the theory of it, I just have problems reducing it to one equation, which is what I wanted explaining. Mr F says: It would have been helpful to say that in the first place. Then you might have got precisely the explanation you apparently needed much sooner.

    I know you won't care, Mr F says: To coin a quote, "I really don't think it's your place to tell me" whether or not I care ....

    but the exam went well and this didn't matter in the end 'cause the question I was given had real eigenvalues (which I find easier to get my head around), so hooray! Mr F says: Good for you. I hope your result reflects that. (Not that I care, of course .... )

    Edit: Thread closed.
    Last edited by mr fantastic; August 21st 2009 at 03:32 PM.
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