7.1: Invariant Subspaces

To begin our study, we will look at subspaces $U$ of $V$ that have special properties under an operator $T$ in $\mathcal{L}(V,V)$.

Definition $\PageIndex{1}$: invariant subspace

Let $V$ be a finite-dimensional vector space over $\mathbb{F}$ with $\dim(V)\ge 1$, and let $T\in \mathcal{L}(V,V)$ be an operator in $V$. Then a subspace $U\subset V$ is called an invariant subspace under $T$ if

$$\begin{equation*} Tu \in U \quad \text{for all \(u\in U\).} \end{equation*}$$
That is, $U$ is invariant under $T$ if the image of every vector in $U$ under $T$ remains within $U$. We denote this as $TU = \{ Tu \mid u\in U \} \subset U$.

Example $\PageIndex{1}$

The subspaces $\kernel(T)$ and $\range(T)$ are invariant subspaces under $T$. To see this, let $u\in\kernel(T)$. This means that $Tu=0$. But, since $0\in\kernel(T)$, this implies that $Tu=0\in \kernel(T)$. Similarly, let $u\in \range(T)$. Since $Tv\in \range(T)$ for all $v\in V$, we certainly also have that $Tu \in \range(T)$.

Example $\PageIndex{2}$

Take the linear operator $T:\mathbb{R}^3\to\mathbb{R}^3$ corresponding to the matrix

$$\begin{equation*} \begin{bmatrix} 1&2&0\\ 1&1&0\\0&0&2 \end{bmatrix} \end{equation*}$$

with respect to the basis $(e_1,e_2,e_3)$. Then $\Span(e_1,e_2)$ and $\Span(e_3)$ are both invariant subspaces under $T$.

An important special case of Definition 7.1.1 involves one-dimensional invariant subspaces under an operator $T$ in $\mathcal{L}(V,V)$. If $\dim(U) = 1$, then there exists a nonzero vector $u$ in $V$ such that

$$U = \{ au \mid a \in \mathbb{F} \}.$$

In this case, we must have

$$T u = \lambda u \quad ~\text{for some \(\lambda \in \mathbb{F}\)}.$$

This motivates the definitions of eigenvectors and eigenvalues of a linear operator, as given in the next section.

Contributors

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.