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# Deciding the JNF watch

1. Q: Let A be an 8x8 matrix A over R, and suppose that and (where is the characteristic polynomial, and 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. 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!
2. Anybody have an idea? I'm not sure I will find much assistance on Coursework.Info!
3. Well, you know that every Jordan block has a char poly of the form . 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.
4. (Original post by DFranklin)
Well, you know that every Jordan block has a char poly of the form . 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...)
5. I think so. Don't forget I've never actually studied JNF in anger.

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