Invariant Subspaces

Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$.

Suppose that $T \in \mathcal{L}(V)$ and $V_{1}, \ldots , V_{m}$ are subspaces of $V$ invariant under $T$. Prove that $V_{1}+\cdots +V_{m}$ is invariant under $T$.

Suppose $T \in \mathcal{L}(V)$. Prove that the intersection of every collection of subspaces of $V$ invariant under $T$ is invariant under $T$.

Prove or give a counterexample: If $V$ is finite-dimensional and $U$ is a subspace of $V$ that is invariant under every operator on $V$, then $U = \left\{ 0\right\}$ or $U = V$.

Suppose $T \in \mathcal{L}\left(\mathbb {R}^{2}\right)$ is defined by $T(x, y) = (-3 y, x)$. Find the eigenvalues of $T$.

Define $T \in \mathcal{L}\left(\mathbf{F}^{2}\right)$ by $T(w, z) = (z, w)$. Find all eigenvalues and eigenvectors of $T$.

Define $T \in \mathcal{L}\left(\mathbf{F}^{3}\right)$ by $T\left(z_{1}, z_{2}, z_{3}\right) = \left(2 z_{2}, 0,5 z_{3}\right)$. Find all eigenvalues and eigenvectors of $T$.

Suppose $P \in \mathcal{L}(V)$ is such that $P^{2} = P$. Prove that if $\lambda$ is an eigenvalue of $P$, then $\lambda = 0$ or $\lambda = 1$.

Define $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathcal{P}(\mathbb {R})$ by $T p = p^{\prime }$. Find all eigenvalues and eigenvectors of $T$.

Define $T \in \mathcal{L}\left(\mathcal{P}_{4}(\mathbb {R})\right)$ by $(T p)(x) = x p^{\prime }(x)$ for all $x \in \mathbb {R}$. Find all eigenvalues and eigenvectors of $T$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $\alpha \in \mathbf{F}$. Prove that there exists $\delta > 0$ such that $T-\lambda I$ is invertible for all $\lambda \in \mathbf{F}$ such that $0 < \left|\alpha -\lambda \right| < \delta$.

Suppose $V = U \oplus W$, where $U$ and $W$ are nonzero subspaces of $V$. Define $P \in \mathcal{L}(V)$ by $P(u+w) = u$ for each $u \in U$ and each $w \in W$. Find all eigenvalues and eigenvectors of $P$.

Suppose $T \in \mathcal{L}(V)$. Suppose $S \in \mathcal{L}(V)$ is invertible.

Give an example of an operator on $\mathbb {R}^{4}$ that has no (real) eigenvalues.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $\lambda \in \mathbf{F}$. Show that $\lambda$ is an eigenvalue of $T$ if and only if $\lambda$ is an eigenvalue of the dual operator $T^{\prime } \in \mathcal{L}\left(V^{\prime }\right)$.

Suppose $v_{1}, \ldots , v_{n}$ is a basis of $V$ and $T \in \mathcal{L}(V)$. Prove that if $\lambda$ is an eigenvalue of $T$, then

where $\mathcal{M}(T)_{j, k}$ denotes the entry in row $j$, column $k$ of the matrix of $T$ with respect to the basis $v_{1}, \ldots , v_{n}$.

Suppose $\mathbf{F} = \mathbb {R}, T \in \mathcal{L}(V)$, and $\lambda \in \mathbb {R}$. Prove that $\lambda$ is an eigenvalue of $T$ if and only if $\lambda$ is an eigenvalue of the complexification $T_{\mathrm{C}}$.

Suppose $\mathbf{F} = \mathbb {R}, T \in \mathcal{L}(V)$, and $\lambda \in \mathbb {C}$. Prove that $\lambda$ is an eigenvalue of the complexification $T_{\mathrm{C}}$ if and only if $\bar{\lambda }$ is an eigenvalue of $T_{\mathrm{C}}$.

Show that the forward shift operator $T \in \mathcal{L}\left(\mathbf{F}^{\infty }\right)$ defined by

has no eigenvalues.

Define the backward shift operator $S \in \mathcal{L}\left(\mathbf{F}^{\infty }\right)$ by

Suppose $T \in \mathcal{L}(V)$ is invertible.

Suppose $T \in \mathcal{L}(V)$ and there exist nonzero vectors $u$ and $w$ in $V$ such that

Prove that 3 or -3 is an eigenvalue of $T$.

Suppose $V$ is finite-dimensional and $S, T \in \mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.

Suppose $A$ is an $n$-by- $n$ matrix with entries in $\mathbf{F}$. Define $T \in \mathcal{L}\left(\mathbf{F}^{n}\right)$ by $T x = A x$, where elements of $\mathbf{F}^{n}$ are thought of as $n$-by- 1 column vectors.

Suppose $T \in \mathcal{L}(V)$ and $u, w$ are eigenvectors of $T$ such that $u+w$ is also an eigenvector of $T$. Prove that $u$ and $w$ are eigenvectors of $T$ corresponding to the same eigenvalue.

Suppose $T \in \mathcal{L}(V)$ is such that every nonzero vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity operator.

Suppose that $V$ is finite-dimensional and $k \in \left\{ 1, \ldots , \operatorname {dim} V-1\right\}$. Suppose $T \in \mathcal{L}(V)$ is such that every subspace of $V$ of dimension $k$ is invariant under $T$. Prove that $T$ is a scalar multiple of the identity operator.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that $T$ has at most $1+\operatorname {dim}$ range $T$ distinct eigenvalues.

Suppose $T \in \mathcal{L}\left(\mathbb {R}^{3}\right)$ and $-4,5$, and $\sqrt{7}$ are eigenvalues of $T$. Prove that there exists $x \in \mathbb {R}^{3}$ such that $T x-9 x = (-4,5, \sqrt{7})$.

Suppose $T \in \mathcal{L}(V)$ and $(T-2 I)(T-3 I)(T-4 I) = 0$. Suppose $\lambda$ is an eigenvalue of $T$. Prove that $\lambda = 2$ or $\lambda = 3$ or $\lambda = 4$.

Give an example of $T \in \mathcal{L}\left(\mathbb {R}^{2}\right)$ such that $T^{4} = -I$.

Suppose $T \in \mathcal{L}(V)$ has no eigenvalues and $T^{4} = I$. Prove that $T^{2} = -I$.

Suppose $T \in \mathcal{L}(V)$ and $m$ is a positive integer.

Suppose $V$ is finite-dimensional and $v_{1}, \ldots , v_{m} \in V$. Prove that the list $v_{1}, \ldots , v_{m}$ is linearly independent if and only if there exists $T \in \mathcal{L}(V)$ such that $v_{1}, \ldots , v_{m}$ are eigenvectors of $T$ corresponding to distinct eigenvalues.

Suppose that $\lambda_{1}, \ldots , \lambda_{n}$ is a list of distinct real numbers. Prove that the list $e^{\lambda_{1} x}, \ldots , e^{\lambda_{n} x}$ is linearly independent in the vector space of real-valued functions on $\mathbb {R}$.

Suppose that $\lambda_{1}, \ldots , \lambda_{n}$ is a list of distinct positive numbers. Prove that the list $\cos \left(\lambda_{1} x\right), \ldots , \cos \left(\lambda_{n} x\right)$ is linearly independent in the vector space of real-valued functions on $\mathbb {R}$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Define $\mathcal{A} \in \mathcal{L}(\mathcal{L}(V))$ by

for each $S \in \mathcal{L}(V)$. Prove that the set of eigenvalues of $T$ equals the set of eigenvalues of $\mathcal{A}$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $U$ is a subspace of $V$ invariant under $T$. The quotient operator $T / U \in \mathcal{L}(V / U)$ is defined by

for each $v \in V$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that $T$ has an eigenvalue if and only if there exists a subspace of $V$ of dimension $\operatorname {dim} V-1$ that is invariant under $T$.

Suppose $S, T \in \mathcal{L}(V)$ and $S$ is invertible. Suppose $p \in \mathcal{P}(\mathbf{F})$ is a polynomial. Prove that

Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$ invariant under $T$. Prove that $U$ is invariant under $p(T)$ for every polynomial $p \in \mathcal{P}(\mathbf{F})$.

Define $T \in \mathcal{L}\left(\mathbf{F}^{n}\right)$ by $T\left(x_{1}, x_{2}, x_{3}, \ldots , x_{n}\right) = \left(x_{1}, 2 x_{2}, 3 x_{3}, \ldots , n x_{n}\right)$.

Suppose that $V$ is finite-dimensional, $\operatorname {dim} V > 1$, and $T \in \mathcal{L}(V)$. Prove that $\left\{ p(T): p \in \mathcal{P}(\mathbf{F})\right\} \neq \mathcal{L}(V)$.