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Deciding the JNF

Q: Let A be an 8x8 matrix A over R, and suppose that cA(z)=(1−z)8c_A(z) = (1-z)^8 and μA(z)=(z−1)4\mu _A(z) = (z-1)^4 (where cA(z)c_A(z) is the characteristic polynomial, and μA(z)\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. ∏i=1t(λi−x)ki\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 latex formula:

\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! :frown:
Reply 1
Avatar for Kolya
Kolya
OP
17
Anybody have an idea? I'm not sure I will find much assistance on Coursework.Info!
Reply 2
Avatar for DFranklin
DFranklin
18
Well, you know that every Jordan block has a char poly of the form (1−x)ki(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.
Reply 3
Avatar for Kolya
Kolya
OP
17
DFranklin
Well, you know that every Jordan block has a char poly of the form (1−x)ki(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...)
Reply 4
Avatar for DFranklin
DFranklin
18
I think so. Don't forget I've never actually studied JNF in anger.

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