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Eigenvalues / eigenvectors for squared matrix watch

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    Hey I've been trying to do this question:

    Find the eigenvalues and corresponding eigenvectors of the matrice B^2, where

    \left(\begin{matrix}-5 & -3 & 0 \\ 1 & -8 & 1 \\ -1 & 3 & -6\end{matrix}\right)

    I've tried going about this the normal way [ det (A-λI) = 0 ] then finding the eigenvalues and eigenvectors correspondingly, but it runs out too complicated with huge numbers like 2304 etc. Is there an easier way to go about this?
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    I think the [ det (A-λI) = 0 ] method comes out nice.

    I get eigenvalues -5,-8,-6
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    (Original post by insparato)
    I think the [ det (A-λI) = 0 ] method comes out nice.

    I get eigenvalues -5,-8,-6
    You can always use those special properties (i forget them) which simplify calculating the determinant. I.e. calculate eigenvalues and corresponding eigenvectors of the matrice B, and then there is a formula to find the eigenvalues and corresponding eigenvectors of the matrice B^2.
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    Please, people, for my sanity, the word is matrix. :p:
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    (Original post by generalebriety)
    Please, people, for my sanity, the word is matrix. :p:
    Haha :ditto: :ditto: :ditto:
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    (Original post by coffeym)
    Haha :ditto: :ditto: :ditto:
    Sorry getting mixed up with my plurals there
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    Instead of all of this spamming, did you get the eigenvalues out using the det method?
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    (Original post by insparato)
    Instead of all of this spamming, did you get the eigenvalues out using the det method?
    I don't think it comes out nicely. At least, not without a fight.

    Edit: I got a characteristic equation of t^3 + 75t^2 - 5296t + 83088 = 0. Which, according to some online cubic solver, isn't very nice. So yeah, I made a mistake somewhere. :p: What was your method, Aaron?
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    (Original post by generalebriety)
    I don't think it comes out nicely. At least, not without a fight.
    I don't see any problem with the standard method (although either insparato or I have a sign error for one of the roots).

    Of course, that's just the eigenvalues (and I can't be bothered to do the eigenvectors) for the matrix andrewlee89 provided.

    I'm not sure how that's supposed to be related to a matrix B^2. I'm guessing the matrix supposed to be B and we find need to find eigenvalues for B^2. In which case if v is an eigenvector for B, just apply B twice to see what happens with B^2.
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    \left(\begin{matrix}-5 & -3 & 0 \\ 1 & -8 & 1 \\ -1 & 3 & -6\end{matrix}\right)

     

\left|\begin{matrix}-5-\lambda & -3 & 0 \\ 1 & -8-\lambda & 1 \\ -1 & 3 & -6-\lambda \end{matrix}\right| = 0

     (-5-\lambda)[(-8-\lambda)(-6-\lambda)-3] + 3[(-6-\lambda)+1] = 0

     (-5-\lambda)(-8-\lambda)(-6-\lambda) -3(-5-\lambda) + 3(-5-\lambda) = 0

     (-5-\lambda)(-8-\lambda)(-6-\lambda)  = 0

     \lambda = -5,-8,-6
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    (Original post by insparato)
    \left(\begin{matrix}-5 & -3 & 0 \\ 1 & -8 & 1 \\ -1 & 3 & -6\end{matrix}\right)

     

\left|\begin{matrix}-5-\lambda & -3 & 0 \\ 1 & -8-\lambda & 1 \\ -1 & 3 & -6-\lambda \end{matrix}\right| = 0

     (-5-\lambda)[(-8-\lambda)(-6-\lambda)-3] + 3[(-6-\lambda)+1] = 0

     (-5-\lambda)(-8-\lambda)(-6-\lambda) -3(-5-\lambda) + 3(-5-\lambda) = 0

     (-5-\lambda)(-8-\lambda)(-6-\lambda)  = 0

     \lambda = -5,-8,-6
    But I thought he wanted the eigenvalues of B^2, where the above was B...

    (Original post by DFranklin)
    In which case if v is an eigenvector for B, just apply B twice to see what happens with B^2.
    Good point.
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    Oh, my mind skipped that part! I'll have a go now. Im assuming the matrix given is B ? although that hasnt been said.
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    (Original post by insparato)
    Oh, my mind skipped that part! I'll have a go now. Im assuming the matrix given is B ? although that hasnt been said.
    No, it's poorly worded, but I think that's what he meant.

    DFranklin's idea is probably best.
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    Yeah ive just had ago and looking at what David says i think that would be a better approach. By heck i need to do some matrix algebra.
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    (Original post by generalebriety)
    No, it's poorly worded, but I think that's what he meant.

    DFranklin's idea is probably best.
    Oh my goodness so sorry! Yeah the matrix given was B= \left(\begin{matrix}-5 & -3 & 0 \\ 1 & -8 & 1 \\ -1 & 3 & -6\end{matrix}\right)

    And they want the corresponding eigenvalues and eigenvectors for B^2 . Sorry for the confusion. Any luck with the numbers sprawling into 4 digits ?
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    (Original post by andrewlee89)
    Oh my goodness so sorry! Yeah the matrix given was B= \left(\begin{matrix}-5 & -3 & 0 \\ 1 & -8 & 1 \\ -1 & 3 & -6\end{matrix}\right)

    And they want the corresponding eigenvalues and eigenvectors for B^2 . Sorry for the confusion. Any luck with the numbers sprawling into 4 digits ?
    Please tell me you're not actually working out B^2 and its characteristic polynomial... :eek:
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    (Original post by SsEe)
    Please tell me you're not actually working out B^2 and its characteristic polynomial... :eek:
    What other choice is there? I've been at this for the past two days (okay not yesterday, I took a rest) The problem is there's no answer in my book to this question sigh
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    (Original post by andrewlee89)
    What other choice is there? I've been at this for the past two days (okay not yesterday, I took a rest) The problem is there's no answer in my book to this question sigh
    Suppose Bv = kv (i.e. v is an eigenvector for B with eigenvalue k).

    Can you work out what B^2{\bf v} is? What does this tell you about the eigenvalues and eigenvectors of B^2?
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    Think about what an eigenvector is? There's probably some blurb before you come onto the subject explaining what's special about an eigenvector in a non-mathematical sense.
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    (Original post by DFranklin)
    Suppose Bv = kv (i.e. v is an eigenvector for B with eigenvalue k).

    Can you work out what B^2{\bf v} is? What does this tell you about the eigenvalues and eigenvectors of B^2?
    Hi, I'm new here, hello.

    Sorry to resurrect an old thread, but would it be correct to say that B^2{\bf v} = k^2{\bf v} ? I can't see how to prove this, my matrix algebra is really lacking. Could somebody show me?

    Thanks.
 
 
 
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