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Math Help - Put the following matrices in canonical form?

  1. #1
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    Put the following matrices in canonical form?

    a]
    (1 0 1)
    (0 1 0)
    (0 0 1)

    b]
    (1 0 1)
    (0 1 0)
    (1 0 1)

    c]
    (1 0 0 1)
    (0 1 1 0)
    (0 0 1 0)
    (1 0 0 0)

    Help please.
    Last edited by Ackbeet; February 22nd 2011 at 02:26 AM. Reason: Moved from Differential Equations.
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  2. #2
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    Quote Originally Posted by Borkborkmath View Post
    a]
    (1 0 1)
    (0 1 0)
    (0 0 1)

    b]
    (1 0 1)
    (0 1 0)
    (1 0 1)

    c]
    (1 0 0 1)
    (0 1 1 0)
    (0 0 1 0)
    (1 0 0 0)

    Help please.

    1) What do you call "canonical form" to? Do you mean Jordan Canonical Form, Smith

    Canonical form, Rational Canonical form...?

    2) Show some self work and show where you're stuck.

    Tonio
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  3. #3
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    I'm not sure which canonical form is required, since my book doesn't say. It just tells me to put those matrices into canonical form.

    I know that λI is a canonical form for Jordan matrices and that
    (λ 1 0 0)
    (0 λ 1 0)
    (0 0 λ 1)
    (0 0 0 λ)
    is another canonical form.

    The thing is that I do not know what to do or where to start.
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  4. #4
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    Quote Originally Posted by Borkborkmath View Post
    I'm not sure which canonical form is required, since my book doesn't say. It just tells me to put those matrices into canonical form.

    I know that λI is a canonical form for Jordan matrices and that
    (λ 1 0 0)
    (0 λ 1 0)
    (0 0 λ 1)
    (0 0 0 λ)
    is another canonical form.

    The thing is that I do not know what to do or where to start.


    Well, from what you say it seems (just seems!) that they mean the Jordan Canonical Form (JCF), but

    in order to be completely sure you must read the book before the part of that question.

    And even if it actually is the JCF and you don't know about this, it is too involved a question to write

    down an explanation in this forum. You better go to your instructor.

    Tonio
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  5. #5
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    Since you mentioned the Jordan Canonical form.

    The first thing you need is the characteristic polynomial

    a)

    \displaystyle c(\lambda)=|A-\lambda I|=\begin{vmatrix} 1-\lambda & 0 & 1 \\ 0 & 1-\lambda & 0 \\ 0 & 0 & 1-\lambda \end{vmatrix}=(1-\lambda)^3

    Putting \lambda =1 back into the matrix gives

    \displaystyle (A-I)=\begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

    The dimension of the nullspace of this matrix is 2 so we need to find the Jordan cannonical form.

    Note that the minimum polynomial must be of the form
    m(\lambda)=(1-\lambda)^a where a=1,2,3

    because m(\lambda) | c(\lambda) and have the same roots.

    Since (A-I) \ne 0 matrix the minimum polynomial cannot be (\lambda -1) Now try (\lambda -1)^2=\lambda^2-2\lambda+1 and check that A^2-2A+I=0this gives that a=2
    The degree of the minimum polynomial gives the size of the largest Jordan block. So the JCF has one 2 by 2 block and one 1 by 1 block. So the JCF is

    \displaystyle \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}

    Also do you need the matricies P \text{ and } P^{-1} such that
    P^{-1}AP=J where J is the JCF?

    See if you can do the next one!
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  6. #6
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    Thank you very much for your explanations!
    I hadn't had time to come back and check until now, but I think I got b:
    [0 0 0]
    [0 1 0]
    [0 0 2]?
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