Minimal polynomials and generalized eigenspaces

Hi Professor Kim,
Having read various notes and definitions on the internet, i still don’t understand how to calculate the minimal polynomial in your Linear Algebra course.

If i take the Homework 4 sheet as an example, I have no problem working out the characteristic polynomials in questions 2 and 3, which i find to be (X-1)(X-2)(X-1), (X-2)(X-2)(X-1) [Q.2] and (X-6)^4 [Q.3] … could you tell me how to work out m_A(X) from this?

Also, in the notes it says that V1(lambda) is contained in V2(lambda) and so on… which agrees with my answer to question 2, but in question 3 i have worked out V1, V2 and V3 and they come out as

V1 = (0,-2,-1,1 )^t, V2 = (0,1,1,0 )^t and (0,-1,0,1 )^t V3= (1,0,0)^t and (0,1,0 )^t and (0,0,1)^t

(and the last result means there’s no point in carrying on) And as you can see none of the previous vectors are contained in the next set.

Could you tell me where i’m going wrong? The only thing i can think of is if b=1 (so the mA(X) = (X-6)^1 and we only work out V1) then the containment issue is void.

Thanks in advance for any help.

Reply:

I will make a few comments based on the computation you give, with no guarantee of their correctness. Most importantly, the V_t(lambda) are subspaces, so you should write something like

V_1=Span (0,-2,-1,1 )^t, V_2= Span{(0,1,1,0 )^t, (0,-1,0,1 )^t}, and so on.

This resolves part of your difficulty. That is, the assertion is that

V_1\subset V_2\subset V_3, …

as subspaces. The containment isn’t about any particular bases you may have chosen. For example, note that

(0,-2,-1,1 )^t=(0,-1,0,1 )^t-(0,1,1,0 )^t

so that definitely V_1 \subset V_2. Try working it out again with this in mind.

Having said this, I should point out that what you wrote for V_3 doesn’t make sense, since your vectors should have four components. Read some of the previous posts for some comments on the minimal polynomial.

MK

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