Answer to Question #163676 in Linear Algebra for Nikhil Rawat

Question

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

Answer 1

We seek eigenvalues of this operator:

$T(a,b,c)=(a-b,b,c)=(\lambda a, \lambda b, \lambda c)$.

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

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

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.

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

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

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