# 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

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