Question on minimal polynomial

Dear Prof Kim,

There is an example about minimal polynomials on the online notes for the course 2201 page 20 which I am not sure about it.

I know how to find the characteristic polynomial, (x-2)^3 but I am not clear how to find m(T)=0. Would you please help me ?
Many thanks.

Reply:

Once you have the characteristic polynomial ch(x) , you use the fact that the minimal polynomial m(x) is monic and divides ch(x). In this case, because

ch()=(x-2)^3

this means

m(x)=(x-2)^3, (x-2)^2, or (x-2).

Note that the matrix (T-2) is

0 1 0
0 0 0
0 0 0

(Recall that (T-2) means `T minus 2 times the identity matrix’.)

Of course we know that (T-2)^3=ch(T) must be zero by the Cayley-Hamilton theorem. So we need only check for the possibility that a smaller power might be zero. In fact, it turns out by direct computation that (T-2)^2=0. Of course (T-2) itself is not zero. This tells us that the minimal polynomial is (x-2)^2.

MK

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3 Comments

  1. Posted November 6, 2007 at 11:05 am | Permalink | Reply

    Dear Prof Kim,

    I’m not quite sure how to apply the theorem on the example on the online notes
    page 21. Could you please help me ?

    Many thanks.

    Reply:

    For this, I won’t answer just yet. Think carefully about what I wrote above concerning the previous example and ask again if you’re still stuck.

    MK

  2. Dree
    Posted November 6, 2007 at 12:22 pm | Permalink | Reply

    Dear Prof Kim,

    I think I was confused because I got my (x-2)(x-3) matrix equals zero, but now I think I understood.

    Thanks

  3. Posted November 6, 2007 at 12:40 pm | Permalink | Reply

    If you’d like, you should write out an explanation of the example here in your own words. It could be a good opportunity to check your understanding.

    MK

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