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    Q: Let A be an 8x8 matrix A over R, and suppose that c_A(z) = (1-z)^8 and \mu _A(z) = (z-1)^4 (where c_A(z) is the characteristic polynomial, and \mu _A(z) is the minimal polynomial) Write down the possible JNFs for A. How would you decide which was the correct JNF?

    I am confused about how to approach this. The facts I know are that the minimal polynomial of J is the equal to the product of the characteristic polynomials of the jordan blocks, i.e. \prod ^t_{i=1} (\lambda _i - x)^{k_i} and that the minimal polynomial of J is the least common multiple of minimal polynomials of the Jordan blocks,
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \prod ^t _{i=1} (x - \lambda _i )^{k_{i1}
    . However, I have no idea how to turn these nice-sounding facts (which I don't particularly understand) into an answer for this question!
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    Anybody have an idea? I'm not sure I will find much assistance on Coursework.Info!
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    Well, you know that every Jordan block has a char poly of the form (1-x)^{k_i}. So the lowest common multiple is going to simply be (1-x)^M where M = max{k_i}. From here it's not hard to enumerate the various possibilities.

    I confess I'm not sure I know what they're expecting as the answer to "how would you decide the correct JNF".

    Edit: Zhen's suggestion is one of the possibilites, but there are several more.
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    (Original post by DFranklin)
    Well, you know that every Jordan block has a char poly of the form (1-x)^{k_i}. So the lowest common multiple is going to simply be (1-x)^M where M = max{k_i}. From here it's not hard to enumerate the various possibilities.
    So the answer would just be this list?

    J(L4) + J(L4)
    J(L4) + J(L2) + J_(L2)
    J(L4) + J(L2) + J(L1) + J(L1)
    J(L4) + J(L1) + J(L1) + J(L1) + J(L1)

    where, for example, J(L4) is the jordan block with eigenvalue L, which is 1 here, and size 4x4, and + just stands for the Jordan sum thingy. (To use the technical terminology.)

    (Apologies for the long delay before replying...)
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    I think so. Don't forget I've never actually studied JNF in anger.
 
 
 
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Updated: November 25, 2008

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