Math Problems Vault

Problem 7A.1

Suppose $n$ is a positive integer. Define $T \in \mathcal{L}(\mathbf{F}^{n})$ by

T(z_{1}, \ldots , z_{n}) = (0, z_{1}, \ldots , z_{n-1}).

Find a formula for $T^{*}(z_{1}, \ldots , z_{n})$ .

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Problem 7A.2

Suppose $T \in \mathcal{L}(V, W)$ . Prove that

T = 0 \iff T^{*} = 0 \iff T^{*} T = 0 \iff T T^{*} = 0.

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Problem 7A.3

Suppose $T \in \mathcal{L}(V)$ and $\lambda \in \mathbf{F}$ . Prove that

\lambda \text{ is an eigenvalue of } T \iff \overline{\lambda } \text{ is an eigenvalue of } T^{*}.

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Problem 7A.4

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

U \text{ is invariant under } T \iff U^{\perp } \text{ is invariant under } T^{*}.

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Problem 7A.5

Suppose $T \in \mathcal{L}(V, W)$ . Suppose $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$ and $f_{1}, \ldots , f_{m}$ is an orthonormal basis of $W$ . Prove that

\left\| T e_{1}\right\| ^{2}+\cdots +\left\| T e_{n}\right\| ^{2} = \left\| T^{*} f_{1}\right\| ^{2}+\cdots +\left\| T^{*} f_{m}\right\| ^{2}.

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Remark.

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The numbers $\left\| T e_{1}\right\|^{2}, \ldots , \left\| T e_{n}\right\|^{2}$ in the equation above depend on the orthonormal basis $e_{1}, \ldots , e_{n}$ , but the right side does not depend on $e_{1}, \ldots , e_{n}$ . Thus the equation above shows that the sum on the left side does not depend on which orthonormal basis $e_{1}, \ldots , e_{n}$ is used.

Problem 7A.6

(a)(b)

Suppose $T \in \mathcal{L}(V, W)$ . Prove that

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(a)

$T$ is injective $\iff$ $T^{*}$ is surjective;

(b)

$T$ is surjective $\iff$ $T^{*}$ is injective.

Problem 7A.7

(a)(b)

Prove that if $T \in \mathcal{L}(V, W)$ , then

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(a)

$\operatorname {dim} \operatorname {null} T^{*} = \operatorname {dim} \operatorname {null} T + \operatorname {dim} W-\operatorname {dim} V$ ;

(b)

$\operatorname {dim} \operatorname {range} T^{*} = \operatorname {dim} \operatorname {range} T$ .

Problem 7A.8

Suppose $A$ is an $m$ -by- $n$ matrix with entries in $\mathbf{F}$ . Use (b) in Exercise 7A.7 to prove that the row rank of $A$ equals the column rank of $A$ .

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Remark.

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This exercise asks for yet another alternative proof of a result that was previously proved in 3.57 and 3.133.

Theorems 3.57 and 3.133 (column rank equals row rank): suppose $A \in \mathbf{F}^{m, n}$ . Then the column rank of $A$ equals the row rank of $A$ .

Problem 7A.9

Prove that the product of two self-adjoint operators on $V$ is self-adjoint if and only if the two operators commute.

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Problem 7A.10

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$ . Prove that $T$ is self-adjoint if and only if

\left\langle T v, v \right\rangle = \left\langle T^{*} v, v \right\rangle

for all $v \in V$ .

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Problem 7A.11

(a)(b)(c)

Define an operator $S: \mathbf{F}^{2} \rightarrow \mathbf{F}^{2}$ by $S(w, z) = (-z, w)$ .

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(a)

Find a formula for $S^{*}$ .

(b)

Show that $S$ is normal but not self-adjoint.

(c)

Find all eigenvalues of $S$ .

If $\mathbf{F} = \mathbb {R}$ , then $S$ is the operator on $\mathbb {R}^{2}$ of counterclockwise rotation by $90^{\circ }$ .

Problem 7A.12

An operator $B \in \mathcal{L}(V)$ is called skew if

B^{*} = -B.

Suppose that $T \in \mathcal{L}(V)$ . Prove that $T$ is normal if and only if there exist commuting operators $A$ and $B$ such that $A$ is self-adjoint, $B$ is a skew operator, and $T = A+B$ .

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Problem 7A.13

(a)(b)

Suppose $\mathbf{F} = \mathbb {R}$ . Define $\mathcal{A} \in \mathcal{L}(\mathcal{L}(V))$ by $\mathcal{A} T = T^{*}$ for all $T \in \mathcal{L}(V)$ .

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(a)

Find all eigenvalues of $\mathcal{A}$ .

(b)

Find the minimal polynomial of $\mathcal{A}$ .

Problem 7A.14

(a)(b)

Define an inner product on $\mathcal{P}_{2}(\mathbb {R})$ by $\left\langle p, q \right\rangle = \int_{0}^{1} p q$ . Define an operator $T \in \mathcal{L}(\mathcal{P}_{2}(\mathbb {R}))$ by

T(a x^{2}+b x+c) = b x.

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(a)

Show that with this inner product, the operator $T$ is not self-adjoint.

(b)

The matrix of $T$ with respect to the basis $1, x, x^{2}$ is

\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}.

This matrix equals its conjugate transpose, even though $T$ is not self-adjoint. Explain why this is not a contradiction.

Problem 7A.15

(a)(b)

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

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(a)

$T$ is self-adjoint $\iff$ $T^{-1}$ is self-adjoint;

(b)

$T$ is normal $\iff$ $T^{-1}$ is normal.

Problem 7A.16

(a)(b)

Suppose $\mathbf{F} = \mathbb {R}$ .

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(a)

Show that the set of self-adjoint operators on $V$ is a subspace of $\mathcal{L}(V)$ .

(b)

What is the dimension of the subspace of $\mathcal{L}(V)$ in (a) (in terms of $\operatorname {dim} V$ )?

Problem 7A.17

Suppose $\mathbf{F} = \mathbb {C}$ . Show that the set of self-adjoint operators on $V$ is not a subspace of $\mathcal{L}(V)$ .

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Problem 7A.18

Suppose $\operatorname {dim} V \geq 2$ . Show that the set of normal operators on $V$ is not a subspace of $\mathcal{L}(V)$ .

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Problem 7A.19

Suppose $T \in \mathcal{L}(V)$ and $\left\| T^{*} v\right\| \leq \left\| T v\right\|$ for every $v \in V$ . Prove that $T$ is normal.

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Remark.

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This exercise fails on infinite-dimensional inner product spaces, leading to what are called hyponormal operators, which have a well-developed theory.

Problem 7A.20

Suppose $P \in \mathcal{L}(V)$ is such that $P^{2} = P$ . Prove that the following are equivalent.

$P$ is self-adjoint.
$P$ is normal.
There is a subspace $U$ of $V$ such that $P = P_{U}$ .

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Problem 7A.21

Suppose $D: \mathcal{P}_{8}(\mathbb {R}) \rightarrow \mathcal{P}_{8}(\mathbb {R})$ is the differentiation operator defined by $D p = p^{\prime }$ . Prove that there does not exist an inner product on $\mathcal{P}_{8}(\mathbb {R})$ that makes $D$ a normal operator.

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Problem 7A.22

Give an example of an operator $T \in \mathcal{L}(\mathbb {R}^{3})$ such that $T$ is normal but not self-adjoint.

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Problem 7A.23

Suppose $T$ is a normal operator on $V$ . Suppose also that $v, w \in V$ satisfy the equations

\left\| v\right\| = \left\| w\right\| = 2, \quad T v = 3 v, \quad T w = 4 w.

Show that $\left\| T(v+w)\right\| = 10$ .

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Problem 7A.24

Suppose $T \in \mathcal{L}(V)$ and

a_{0}+a_{1} z+a_{2} z^{2}+\cdots +a_{m-1} z^{m-1}+z^{m}

is the minimal polynomial of $T$ . Prove that the minimal polynomial of $T^{*}$ is

\overline{a_{0}}+\overline{a_{1}} z+\overline{a_{2}} z^{2}+\cdots +\overline{a_{m-1}} z^{m-1}+z^{m}.

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Remark.

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This exercise shows that the minimal polynomial of $T^{*}$ equals the minimal polynomial of $T$ if $\mathbf{F} = \mathbb {R}$ .

Problem 7A.25

Suppose $T \in \mathcal{L}(V)$ . Prove that $T$ is diagonalizable if and only if $T^{*}$ is diagonalizable.

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Problem 7A.26

(a)(b)

Fix $u, x \in V$ . Define $T \in \mathcal{L}(V)$ by $T v = \left\langle v, u \right\rangle x$ for every $v \in V$ .

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(a)

Prove that if $V$ is a real vector space, then $T$ is self-adjoint if and only if the list $u, x$ is linearly dependent.

(b)

Prove that $T$ is normal if and only if the list $u, x$ is linearly dependent.

Problem 7A.27

Suppose $T \in \mathcal{L}(V)$ is normal. Prove that

\operatorname {null} T^{k} = \operatorname {null} T \quad \text{ and } \quad \operatorname {range} T^{k} = \operatorname {range} T

for every positive integer $k$ .

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Problem 7A.28

Suppose $T \in \mathcal{L}(V)$ is normal. Prove that if $\lambda \in \mathbf{F}$ , then the minimal polynomial of $T$ is not a polynomial multiple of $(x-\lambda )^{2}$ .

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Problem 7A.29

Prove or give a counterexample: If $T \in \mathcal{L}(V)$ and there is an orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$ such that $\left\| T e_{k}\right\| = \left\| T^{*} e_{k}\right\|$ for each $k = 1, \ldots , n$ , then $T$ is normal.

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Problem 7A.30

Suppose that $T \in \mathcal{L}(\mathbf{F}^{3})$ is normal and $T(1,1,1) = (2,2,2)$ . Suppose $(z_{1}, z_{2}, z_{3}) \in \operatorname {null} T$ . Prove that $z_{1}+z_{2}+z_{3} = 0$ .

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Problem 7A.31

(a)(b)

Fix a positive integer $n$ . In the inner product space of continuous real-valued functions on $[-\pi , \pi ]$ with inner product $\left\langle f, g \right\rangle = \int_{-\pi }^{\pi } f g$ , let

V = \operatorname {span}(1, \cos x, \cos 2 x, \ldots , \cos n x, \sin x, \sin 2 x, \ldots , \sin n x).

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(a)

Define $D \in \mathcal{L}(V)$ by $D f = f^{\prime }$ . Show that $D^{*} = -D$ . Conclude that $D$ is normal but not self-adjoint.

(b)

Define $T \in \mathcal{L}(V)$ by $T f = f^{\prime \prime }$ . Show that $T$ is self-adjoint.

Problem 7A.32

Suppose $T: V \rightarrow W$ is a linear map. Show that under the standard identification of $V$ with $V^{\prime }$ (see 6.58) and the corresponding identification of $W$ with $W^{\prime }$ , the adjoint map $T^{*}: W \rightarrow V$ corresponds to the dual map $T^{\prime }: W^{\prime } \rightarrow V^{\prime }$ . More precisely, show that

T^{\prime }(\varphi _{w}) = \varphi _{T^{*} w}

for all $w \in W$ , where $\varphi_{w}$ and $\varphi_{T^{*} w}$ are defined as in 6.58.

6.58 (Riesz representation theorem, revisited): suppose $V$ is finite-dimensional. For each $v \in V$ , define $\varphi_{v} \in V^{\prime }$ by $\varphi_{v}(u) = \left\langle u, v \right\rangle$ for each $u \in V$ . Then $v \mapsto \varphi_{v}$ is a one-to-one function from $V$ onto $V^{\prime }$ .

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