📂Linear Algebra

Cyclic Subspaces of Vector Spaces

Definition¹

Let us consider $T : V \to V$ as a linear transformation on the vector space $V$. Suppose $\mathbf{v} \ne \mathbf{0} \in V$. The following subspace

$$W = \span\left( \left\{ \mathbf{v}, T\mathbf{v}, T^{2}\mathbf{v}, \dots \right\} \right)$$

is called the $V$ $T$-cyclic subspace generated by $\mathbf{v}$.

Description

The $T$-cyclic subspace is trivially a $T$-invariant subspace. Also, it is the smallest $T$-invariant subspace that includes $\mathbf{v}$.

Theorem¹

Let $T : V \to V$ be a linear transformation on the finite-dimensional vector space $V$. Let us define $W$ as the $T$-cyclic subspace generated by $\mathbf{v} \ne \mathbf{0} \in V$. Suppose $k = \dim(W)$. Then,

1. $\left\{ \mathbf{v}, T\mathbf{v}, \dots, T^{k-1}\mathbf{v} \right\}$ is a basis of $W$.

2. If $a_{0}\mathbf{v} + a_{1}T \mathbf{v} + \cdots + a_{k-1}T^{k-1} \mathbf{v} + T^{k}\mathbf{v} = \mathbf{0}$, then the characteristic polynomial of the restriction $T|_{W}$ is $$f(t) = (-1)^{k}\left( a_{0} + a_{1}t + \cdots +a_{k-1}t^{k-1} + t^{k} \right)$$

Proof

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See Also

• Subspace
• Invariant Subspace