Eigenvalues and Eigenvectors

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 $\left\{ p(T): p \in \mathcal{P}(\mathbf{F})\right\} \neq \mathcal{L}(V)$.

Suppose $T \in \mathcal{L}(V)$. Prove that $9$ is an eigenvalue of $T^{2}$ if and only if $3$ or $-3$ is an eigenvalue of $T$.

Suppose $V$ is a complex vector space and $T \in \mathcal{L}(V)$ has no eigenvalues. Prove that every subspace of $V$ invariant under $T$ is either $\left\{ 0\right\}$ or infinite-dimensional.

Suppose $n$ is a positive integer and $T \in \mathcal{L}\left(\mathbf{F}^{n}\right)$ is defined by

Suppose $\mathbf{F} = \mathbb {C}, T \in \mathcal{L}(V), p \in \mathcal{P}(\mathbb {C})$, and $\alpha \in \mathbb {C}$. Prove that $\alpha$ is an eigenvalue of $p(T)$ if and only if $\alpha = p(\lambda )$ for some eigenvalue $\lambda$ of $T$.

Give an example of an operator on $\mathbb {R}^{2}$ that shows the result in Exercise 4 does not hold if $\mathbb {C}$ is replaced with $\mathbb {R}$.

Exercise 4 (Exercise 5B.4): if $\mathbf{F} = \mathbb {C}$, $T \in \mathcal{L}(V)$, $p \in \mathcal{P}(\mathbb {C})$, and $\alpha \in \mathbb {C}$, then $\alpha$ is an eigenvalue of $p(T)$ if and only if $\alpha = p(\lambda )$ for some eigenvalue $\lambda$ of $T$.

Suppose $T \in \mathcal{L}\left(\mathbf{F}^{2}\right)$ is defined by $T(w, z) = (-z, w)$. Find the minimal polynomial of $T$.

Suppose $T \in \mathcal{L}\left(\mathbb {R}^{2}\right)$ is the operator of counterclockwise rotation by $1^{\circ }$. Find the minimal polynomial of $T$.

Suppose $T \in \mathcal{L}(V)$ is such that with respect to some basis of $V$, all entries of the matrix of $T$ are rational numbers. Explain why all coefficients of the minimal polynomial of $T$ are rational numbers.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $v \in V$. Prove that

Suppose $V$ is a two-dimensional vector space, $T \in \mathcal{L}(V)$, and the matrix of $T$ with respect to some basis of $V$ is

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

Suppose $T \in \mathcal{L}(V)$ and $p \in \mathcal{P}(\mathbf{F})$. Prove that there exists a unique $r \in \mathcal{P}(\mathbf{F})$ such that $p(T) = r(T)$ and $\operatorname {deg} r$ is less than the degree of the minimal polynomial of $T$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ has minimal polynomial $\mu_T(z) = 4+5 z-6 z^{2}-7 z^{3}+2 z^{4}+z^{5}$. Find the minimal polynomial of $T^{-1}$.

Suppose $V$ is a finite-dimensional complex vector space with $\operatorname {dim} V > 0$ and $T \in \mathcal{L}(V)$. Define $f: \mathbb {C} \rightarrow \mathbb {R}$ by

Prove that $f$ is not a continuous function.

Suppose $a_{0}, \ldots , a_{n-1} \in \mathbf{F}$. Let $T$ be the operator on $\mathbf{F}^{n}$ whose matrix (with respect to the standard basis) is

Here all entries of the matrix are 0 except for all 1's on the line under the diagonal and the entries in the last column (some of which might also be 0). Show that the minimal polynomial of $T$ is the polynomial

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $\mu_T$ is the minimal polynomial of $T$. Suppose $\lambda \in \mathbf{F}$. Show that the minimal polynomial of $T-\lambda I$ is the polynomial $q$ defined by $q(z) = \mu_{T}(z+\lambda )$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $p$ is the minimal polynomial of $T$. Suppose $\lambda \in \mathbf{F} \backslash \left\{ 0\right\}$. Show that the minimal polynomial of $\lambda T$ is the polynomial $q$ defined by $q(z) = \lambda^{\operatorname {deg} p} p\left(\frac{z}{\lambda }\right)$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Let $\mathcal{E}$ be the subspace of $\mathcal{L}(V)$ defined by

Prove that $\operatorname {dim} \mathcal{E}$ equals the degree of the minimal polynomial of $T$.

Suppose $T \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ is such that the eigenvalues of $T$ are 3,5,8. Prove that

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

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that $T$ is invertible if and only if $I \in \operatorname {span}\left(T, T^{2}, \ldots , T^{\operatorname {dim} V}\right)$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Let $n = \operatorname {dim} V$. Prove that if $v \in V$, then $\operatorname {span}\left(v, T v, \ldots , T^{n-1} v\right)$ is invariant under $T$.

Suppose $V$ is a finite-dimensional complex vector space. Let $n = \operatorname {dim} V$. Suppose $T \in \mathcal{L}(V)$ is such that 5 and 6 are eigenvalues of $T$ and that $T$ has no other eigenvalues. Prove that $(T-5 I)^{n-1}(T-6 I)^{n-1} = 0$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $U$ is a subspace of $V$ that is invariant under $T$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $U$ is a subspace of $V$ that is invariant under $T$. Prove that the set of eigenvalues of $T$ equals the union of the set of eigenvalues of $\left.T\right|_{U}$ and the set of eigenvalues of $T / U$.

Suppose $\mathbf{F} = \mathbb {R}, V$ is finite-dimensional, and $T \in \mathcal{L}(V)$. Prove that the minimal polynomial of $T_{\mathrm{C}}$ equals the minimal polynomial of $T$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that the minimal polynomial of $T^{\prime } \in \mathcal{L}\left(V^{\prime }\right)$ equals the minimal polynomial of $T$.

Show that every operator on a finite-dimensional vector space of dimension at least two has an invariant subspace of dimension two.

Prove or give a counterexample: If $T \in \mathcal{L}(V)$ and $T^{2}$ has an upper-triangular matrix with respect to some basis of $V$, then $T$ has an upper-triangular matrix with respect to some basis of $V$.

Suppose $A$ and $B$ are upper-triangular matrices of the same size, with $\alpha_{1}, \ldots , \alpha_{n}$ on the diagonal of $A$ and $\beta_{1}, \ldots , \beta_{n}$ on the diagonal of $B$.

Suppose $T \in \mathcal{L}(V)$ is invertible and $v_{1}, \ldots , v_{n}$ is a basis of $V$ with respect to which the matrix of $T$ is upper triangular, with $\lambda_{1}, \ldots , \lambda_{n}$ on the diagonal. Show that the matrix of $T^{-1}$ is also upper triangular with respect to the basis $v_{1}, \ldots , v_{n}$, with

on the diagonal.

Give an example of an operator whose matrix with respect to some basis contains only 0 's on the diagonal, but the operator is invertible.

Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagonal, but the operator is not invertible.

Suppose $\mathbf{F} = \mathbb {C}, V$ is finite-dimensional, and $T \in \mathcal{L}(V)$. Prove that if $k \in \left\{ 1, \ldots , \operatorname {dim} V\right\}$, then $V$ has a $k$-dimensional subspace invariant under $T$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $v \in V$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and there exists a nonzero vector $v \in V$ such that $T^{2} v+2 T v = -2 v$.

Suppose $B$ is a square matrix with complex entries. Prove that there exists an invertible square matrix $A$ with complex entries such that $A^{-1} B A$ is an upper-triangular matrix.

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$. Show that the following are equivalent.

Suppose $\mathbf{F} = \mathbb {C}$ and $V$ is finite-dimensional. Prove that if $T \in \mathcal{L}(V)$, then there exists a basis of $V$ with respect to which $T$ has a lower-triangular matrix.