Linear Algebra - Jordan form basis

  1. Hi all,
    I'm having trouble finding jordan basis for matrix A, e.g. the P matrix of: [itex]J=P^{-1}AP[/itex]
    Given [itex]A = \begin{pmatrix} 4 & 1 & 1 & 1 \\ -1 & 2 & -1 & -1 \\ 6 & 1 & -1 & 1 \\ -6 & -1 & 4 & 2 \end{pmatrix}[/itex]

    I found Jordan form to be: [itex]J = \begin{pmatrix} -2 & & & \\ & 3 & 1 & \\ & & 3 & \\ & & & 3 \end{pmatrix}[/itex]

    Now wer'e looking for [itex]v_1, v_2, v_3, v_4[/itex] such that:

    [itex] Av_1 = -2v_1 → (A+2I)v_1=0[/itex]
    [itex]Av_2 = 3v_2 → (A-3I)v_2=0[/itex]
    [itex]Av_3 = v_2+3v_3 → (A-3I)v_3=v_2[/itex]
    [itex]Av_4 = 3v_4 → (A-3I)v_4=0 [/itex]

    So now I find: [itex]v_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ -1 \end{pmatrix} \hspace{10mm} v_2,v_4 = \begin{pmatrix} 1 \\ 0 \\ 1 \\ -2 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \\ 0 \\ -1 \end{pmatrix}[/itex]

    Now I try to solve [itex] (A-3I)v_3=v_2[/itex] for each of the possible v2's I just found above, but there's no solution for any of em'...

    [itex]A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 \\ 6 & 1 & -4 & 1 \\ -6 & -1 & 4 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}=\begin{pmatrix} 1 \\ 0 \\ 1 \\ -2 \end{pmatrix}\hspace{5mm} OR \hspace{5mm} A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 \\ 6 & 1 & -4 & 1 \\ -6 & -1 & 4 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}=\begin{pmatrix} 0 \\ 1 \\ 0 \\ -1 \end{pmatrix}[/itex]


    Where am I going wrong? Thanks in advance!
     
    Last edited: Jun 29, 2012
  2. jcsd
  3. I like Serena

    I like Serena 6,192
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    Hi oferon! :smile:

    Did you consider that the proper v2 could be a linear combination of your current v2 and v4?
    What if you try ##\lambda v_2 + \mu v_4## to find v3?
     
  4. If it was a linear combination of other vectors then V1-4 would not be a basis.. Am I wrong?

    Plus, another student told me the method I tried was completly wrong and that the correct method is finding more vectors through
    [itex] Ker (A-λI)^j[/itex] where j=2,3,... depends on how many more vectors I need for my basis.

    Which of the methods should I use? Any why? I'm lost :confused:
     
    Last edited: Jul 10, 2012
  5. I like Serena

    I like Serena 6,192
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    You need to find a ##v_3## that satisfies ##(A-3I)v_3=λv_2+μv_4##.
    When you find it, v1-v4 will form a basis.


    That would work too, but it seems to me that it is a lot more work.
    (Short story: that student is wrong. Your method is fine. You just did not finish it.)


    If you're wondering... try both?
     
  6. Hi, thanks for your kind replies.

    Ok, first I try what you suggested.. I take [itex] (A-3I)v_3 = λv_2+μv_4 [/itex] I get:

    [itex]\begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 \\ 6 & 1 & -4 & 1 \\ -6 & -1 & 4 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}=\begin{pmatrix} λ \\ μ \\ λ \\ -2λ-μ \end{pmatrix} ----> \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 6 & 1 & -4 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}=\begin{pmatrix} λ \\ λ+μ \\ λ \\ -λ-μ \end{pmatrix}[/itex]

    Now I see it must satisfy [itex] μ = -λ[/itex] so I pick [itex]λ=1, μ=-1[/itex] thus [itex] v_2-v_4=\begin{pmatrix} 1 \\ -1 \\ 1 \\ -1 \end{pmatrix}[/itex] so now I solve: [itex]\begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 \\ 6 & 1 & -4 & 1 \\ -6 & -1 & 4 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}=\begin{pmatrix} 1 \\ -1 \\ 1 \\ -1 \end{pmatrix}[/itex]
    But the solutions I get are exactly [itex] \begin{pmatrix} 1 \\ 0 \\ 1 \\ -2 \end{pmatrix} , \begin{pmatrix} 0 \\ 1 \\ 0 \\ -1 \end{pmatrix} [/itex] The same v2,v4... So where am I wrong now?


    Second thing, I've searched all over the net, and found this method. Yet the method the other student told me is what was taught in class. Can I be 100% sure both methods are equivalent and can be used both in all cases?
    I thank you again for your time.
     
  7. Ok, so I asked our instructor about the second question and yes, both methods are good.
    I prefer "my" method, but as you can see I still get stucked with it.. So how do I move on with this [itex] (A-3I)v_3 = λv_2+μv_4 [/itex] ?
    Thanks again
     
  8. I like Serena

    I like Serena 6,192
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    Can you find a 3rd solution that is independent of v2 and v4?
    (Let's say with the first 2 entries set to zero. ;)
     
  9. Hmm, ok I see what you say..
    So now I have 3 final questions to close this case for good:

    1) I thought all solutions were given by span of [itex] \begin{pmatrix} 1 \\ 0 \\ 1 \\ -2 \end{pmatrix} , \begin{pmatrix} 0 \\ 1 \\ 0 \\ -1 \end{pmatrix} [/itex]
    So where did this [itex] \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}[/itex] (tho I agree it IS a solution for this system) come from??

    2) How is it possible that [itex]v_2 , v_4[/itex] are solutions of both homogeneous and non-homogeneous
    [itex](A−3I)v_3=0[/itex] and [itex](A−3I)v_3=v_2-v_4[/itex]. I doubt if it was just by accident..

    3) Final question is how come I'm allowed to go from the equation I got by comparing columns of PJ and AP: [itex](A−3I)v_3=v_2[/itex], ,
    to the equation [itex](A−3I)v_3=λv_2+μv_4[/itex]?
    The third column in J matrix [itex] \begin{pmatrix} 0 \\ 1 \\ 3 \\ 0 \end{pmatrix}[/itex] clearly shows I should find [itex]Av_3=v_2+3v_3[/itex] , not [itex]Av_3=v_2+3v_3-v_4[/itex]

    I appreciate your help alot! Thank you.
     
  10. Oh ok, I discard my 3rd question... The answer is that I pick v2 to be [itex]\begin{pmatrix} 1 \\ -1 \\ 1 \\ -1 \end{pmatrix}[/itex]

    Now I remain only with questions 1, and 2.. More related to equations system rather than J form I suppose
     
  11. OK, please discard all of my question, I'm an idiot :)
    Everything is clear now, I thank you very much for the last time :)
     
  12. I like Serena

    I like Serena 6,192
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    Okay... I just got around to looking at your thread again.
    But it seems you've already answered your own questions.

    Good! :smile:
     
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