7.1: Invariant Subspaces
- Page ID
- 251
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.