Cyclic Subspaces

The smallest subspace comprised of compositions of transform on the same vector.

Definition

1. For a non-zero Vector $v$
2. For the smallest $k$ that allows for $\{v,T(v),T^2(v),\dots ,T^k(v)\}$
3. The cyclic subspace is $C_v=\{v,T(v),T^{k-1}(v)\}$

Alternate Definition

• With $\alpha \in V$ and $p$ be the T-Annihilator of $\alpha$
• Let $deg(p)=k$ Then,

1. $\{\alpha ,T(\alpha ),\dots ,T^{k-1}\alpha \}$ is $T$ Invariant
2. $span\{\alpha ,T\alpha ,\dots ,T^{k-1}\alpha \}$ is T Invariant The set $Z({\alpha }_{j}T)=span\{\alpha ,T\alpha ,\dots ,T^{k-1}\alpha \}$ is the cyclic subspace of $\alpha$

Theorems

• Cyclic Decomposition Theorem
• Cyclic Subspace Corrolary
• Matrix Definition of Cyclic Subspace

Proof

Proving Independence

• Suppose for sake on contradiction that $\{\alpha ,T\alpha ,\dots ,T^{k-1}\alpha \}$ is Linearly Dependent
• Then, there exists a non-trivial linear combination equal to zero ($a_{k-1}{T}^{k-1}\alpha +\cdots +a_{0}a=0$)
• let $p(x)-={\alpha }_{k-1}{x}^{k-1}+\cdots +{\alpha }_0$, then $p(T)\alpha =0$
• This contradicts that $p$ being the $T$ annihilator of $\alpha$ as it also annihilates up to $k$, thus $\{\alpha ,T\alpha ,\dots ,T^{k-1}\alpha \}$ is indep

Proving $T$ invariant

• As $T^j\alpha \in W$ for all $j<k$
• We need to show $T^k\alpha \in span\{\alpha ,T\alpha ,\dots ,T^{k-1}\alpha \}$
• As $p(T)\alpha =0$ then, $T^k\alpha +{\alpha }_{k-1}{T}^{k-1}\alpha \cdots +{\alpha }_{0}I\alpha =0$
• $⟹T^k\alpha \in span\{\alpha ,\dots ,T^{k-1}\alpha \}$
• Thus, $\{\alpha ,\dots ,T^{k-1}\alpha \}$ is $T$-Invariant