Math Problems Vault

Problem 5B.1

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 $T$ .

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Problem 5B.2

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.

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Problem 5B.3

(a)(b)

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

T\left(x_{1}, \ldots , x_{n}\right) = \left(x_{1}+\cdots +x_{n}, \ldots , x_{1}+\cdots +x_{n}\right).

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(a)

Find all eigenvalues and eigenvectors of $T$ .

(b)

Find the minimal polynomial of $T$ .

Remark.

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The matrix of $T$ with respect to the standard basis of $\mathbf{F}^{n}$ consists of all 1 's.

Problem 5B.4

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 $T$ .

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Problem 5B.5

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 $T$ .

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Problem 5B.6

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

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Problem 5B.7

(a)(b)

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(a)

Give an example of $S, T \in \mathcal{L}\left(\mathbf{F}^{2}\right)$ such that the minimal polynomial of $S T$ does not equal the minimal polynomial of $T S$ .

(b)

Suppose $V$ is finite-dimensional and $S, T \in \mathcal{L}(V)$ . Prove that if at least one of $S, T$ is invertible, then the minimal polynomial of $S T$ equals the minimal polynomial of $TS$ .

Problem 5B.8

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

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Remark.

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Because $\operatorname {dim} \mathbb {R}^{2} = 2$ , the degree of the minimal polynomial of $T$ is at most 2. Thus the minimal polynomial of $T$ is not the tempting polynomial $x^{180}+1$ , even though $T^{180} = -I$ .

Problem 5B.9

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.

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Problem 5B.10

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

\operatorname {span}\left(v, T v, \ldots , T^{m} v\right) = \operatorname {span}\left(v, T v, \ldots , T^{\operatorname {dim} V-1} v\right) $$ {#eq-5B.10:Eq} for all integers $m \geq \operatorname {dim} V-1$.

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Problem 5B.11

(a)(b)

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

\begin{pmatrix} a & c \\ b & d \end{pmatrix}

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(a)

Show that $T^{2}-(a+d) T+(a d-b c) I = 0$ .

(b)

Show that the minimal polynomial of $T$ equals

\begin{cases} z-a & \text{ if } b = c = 0 \text{ and } a = d, \\ z^{2}-(a+d) z+(a d-b c) & \text{ otherwise } \end{cases}

Problem 5B.12

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 $T$ .

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Problem 5B.13

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 $T$ .

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Problem 5B.14

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 $T^{-1}$ .

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Problem 5B.15

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

f(\lambda ) = \operatorname {dim} \operatorname {range}(T-\lambda I).

Prove that $f$ is not a continuous function.

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Problem 5B.16

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

\begin{pmatrix} 0 & & & & & -a_{0} \\ 1 & 0 & & & & -a_{1} \\ & 1 & \ddots & & & -a_{2} \\ & & \ddots & & & \vdots \\ & & & & 0 & -a_{n-2} \\ & & & & 1 & -a_{n-1} \end{pmatrix}

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

a_{0}+a_{1} z+\cdots +a_{n-1} z^{n-1}+z^{n}.

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Remark.

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The matrix above is called the companion matrix of the polynomial above. This exercise shows that every monic polynomial is the minimal polynomial of some operator. Hence a formula or an algorithm that could produce exact eigenvalues for each operator on each $\mathbf{F}^{n}$ could then produce exact zeros for each polynomial [by 5.27(a)]. Thus there is no such formula or algorithm. However, efficient numeric methods exist for obtaining very good approximations for the eigenvalues of an operator.

Theorem 5.27(a): suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ ; then the zeros of the minimal polynomial of $T$ are the eigenvalues of $T$ .

Problem 5B.17

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 $q(z) = \mu_{T}(z+\lambda )$ .

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Problem 5B.18

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 $q(z) = \lambda^{\operatorname {deg} p} p\left(\frac{z}{\lambda }\right)$ .

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Problem 5B.19

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

\mathcal{E} = \left\{ q(T): q \in \mathcal{P}(\mathbf{F})\right\} .

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

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Problem 5B.20

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

(T-3 I)^{2}(T-5 I)^{2}(T-8 I)^{2} = 0

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Problem 5B.21

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

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Remark.

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If $\operatorname {dim}\operatorname {range}T + 1<\operatorname {dim} V$ , then this exercise gives a better upper bound than 5.22 for the degree of the minimal polynomial of $T$ . Theorem 5.22 (existence, uniqueness, and degree of minimal polynomial): suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ . Then there is a unique monic polynomial $p \in \mathcal{P}(\mathbf{F})$ of smallest degree such that $p(T) = 0$ . Furthermore, $\operatorname {deg} p \leq \operatorname {dim} V$ .

Problem 5B.22

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

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Problem 5B.23

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 $T$ .

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Problem 5B.24

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 $(T-5 I)^{n-1}(T-6 I)^{n-1} = 0$ .

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Problem 5B.25

(a)(b)

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

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(a)

Prove that the minimal polynomial of $T$ is a polynomial multiple of the minimal polynomial of the quotient operator $T / U$ .

(b)

Prove that

\left(\text{minimal polynomial of } T|_{U}\right) \times \left(\text{minimal polynomial of } T / U\right)

is a polynomial multiple of the minimal polynomial of $T$ .

Remark.

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The quotient operator $T/U$ was defined in Exercise 38 in Section 5A: 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 $(T/U)(v+U) = Tv+U$ for each $v \in V$ .

Problem 5B.26

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 $T / U$ .

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Problem 5B.27

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 $T$ .

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Remark.

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The complexification $T_{\mathrm{C}}$ was defined in 3B.33: suppose $V$ is a real vector space and $T \in \mathcal{L}(V)$ ; the complexification $T_{\mathrm{C}}: V_{\mathrm{C}} \rightarrow V_{\mathrm{C}}$ is defined by $T_{\mathrm{C}}(u+iv) = Tu+iTv$ for all $u, v \in V$ , where $V_{\mathrm{C}}$ is the complexification of $V$ from Exercise 8 in Section 1B.

Problem 5B.28

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 $T$ .

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Remark.

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The dual map $T^{\prime }$ was defined in Section 3F. Definition 3.118 (dual map, $T'$ ): suppose $T \in \mathcal{L}(V, W)$ . The dual map of $T$ is the linear map $T' \in \mathcal{L}(W', V')$ defined for each $\varphi \in W'$ by $T'(\varphi ) = \varphi \circ T$ .

Problem 5B.29

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

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Remark.

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Exercise 6 in Section 5C will give an improvement of this result when $\mathbf{F} = \mathbb {C}$ .