Question #163676

Find the minimal polynomial of T: R^3→R^3 defined by T(a,b,c)= (a-b,b,c)


1
Expert's answer
2021-02-24T07:01:00-0500

We seek eigenvalues of this operator:

T(a,b,c)=(ab,b,c)=(λa,λb,λc)T(a,b,c)=(a-b,b,c)=(\lambda a, \lambda b, \lambda c).

This implies that (λ1)b=(λ1)c=0(\lambda-1)b=(\lambda-1)c=0 and (λ1)a=b(\lambda-1)a=-b.

The only value of λ\lambda which provides a non-zero solution is λ=1\lambda=1.

Therefore, the characteristic polynomial of T is (x-1)3.

The minimal polynomial is a factor of the characteristic polynomial (x-1)3.

It may be (x-1)3, (x-1)2 or x-1. Check them all.

(TE)(a,b,c)=(b,0,0)(T-E)(a,b,c)=(-b,0,0), T-E is not a zero operator.

(TE)2(a,b,c)=(TE)(b,0,0)=(0,0,0)(T-E)^2(a,b,c)=(T-E)(-b,0,0)=(0,0,0), hence, (T-E)2=0.

Therefore, the minimal polynomial of T is (x-1)2.


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