Invariant Subspaces

Let $\mathbf{T}$ be an arbitrary linear operator on a vector space $\mathcal{V}$.

Describe all of the subspaces that are invariant under the identity operator $\mathbf{I}$ on a space $\mathcal{V}$.

Let $\mathbf{T}$ be the linear operator on $\mathbb {R}^{4}$ defined by

and let $\mathcal{X} = \operatorname {span}\left\{ \overrightarrow {e}_1, \overrightarrow {e}_2\right\}$ be the subspace that is spanned by the first two unit vectors in $\mathbb {R}^{4}$.

Let $\mathbf{T}$ and $\mathbf{Q}$ be the matrices

Let $\mathbf{T}$ be a linear operator on a space $\mathcal{V}$, and suppose that

is a basis for $\mathcal{V}$ such that $[\mathbf{T}]_{\mathcal{B}}$ has the block-diagonal form

Explain why $\mathcal{U} = \operatorname {span}\left\{ \overrightarrow {u}_1, \ldots , \overrightarrow {u}_r\right\}$ and $\mathcal{W} = \operatorname {span}\left\{ \overrightarrow {w}_1, \ldots , \overrightarrow {w}_q\right\}$ must each be invariant subspaces under $\mathbf{T}$.

If $\mathbf{T}_{n \times n}$ and $\mathbf{P}_{n \times n}$ are matrices such that

explain why $\mathcal{U} = \operatorname {span}\left\{ \mathbf{P}_{*1}, \ldots , \mathbf{P}_{*r}\right\}$ and $\mathcal{W} = \operatorname {span}\left\{ \mathbf{P}_{*r+1}, \ldots , \mathbf{P}_{*n}\right\}$ are each invariant subspaces under $\mathbf{T}$.

Recall (Example 4.8.3): if $\mathbf{T}$ is regarded as the linear operator on $\mathbb {R}^{n \times 1}$ defined by matrix multiplication $\mathbf{T}(\overrightarrow {x}) = \mathbf{T}\overrightarrow {x}$, then with respect to the standard basis $\mathcal{S}$, $[\mathbf{T}]_{\mathcal{S}} = \mathbf{T}$.

If $\mathbf{A}$ is an $n \times n$ matrix and $\lambda$ is a scalar such that $(\mathbf{A} - \lambda \mathbf{I})$ is singular (i.e., $\lambda$ is an eigenvalue), explain why the associated space of eigenvectors $N(\mathbf{A} - \lambda \mathbf{I})$ is an invariant subspace under $\mathbf{A}$.

Consider the matrix $\mathbf{A} = \begin{pmatrix} -9 & 4 \\ -24 & 11 \end{pmatrix}$.