Why are there two non-trivial T-invariant subspaces and why the restriction to T is cyclic?

Question

Given that T is a cyclic operator on real vector space V with generating vector v, and minimal polnyomial $\mu_{T,v}(x) = x^3 - 8$. I see a theorem in my book stating that every T-invariant subspace of $V = <T,v>$ is cyclic, so I can see why the restriction to T of these T-invariant subspaces is cyclic, but I don't know why these 2 non-trivial T-invariant subspaces exist. Any pointers? Also, how can we give explicit generators in terms of T and v for these nontrivial T-invariant subspaces? I am having a hard time with this chapter and any help is appreciated.

Answer 1

The minimal polynomial is reducible: $\mu_T(x) = (x-2)(x^2 + 2x + 4)$. Each divisor of the minimal polynomial gives you a different invariant subspace; for instance, the divisor $x-2$ gives you the invariant subspace $\ker(T-2I)$, which is generated by $(T^2 + 2T + 4I)v$.