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Chapter 7

Commutators and Anticommutators

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7.1 · Chapter Problems

7.2 · Supplementary Problems

Section 7.1Chapter Problems[43/100%]

Problem 7.1.1

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $n \times n$ matrices. Assume that $[A, B] = 0_n$ and $[A, B]_+ = 0_n$ . What can be said about $AB$ and $BA$ ?

(ii)

Let $A$ , $B$ be $n \times n$ matrices. Assume that $A$ is invertible. Assume that $[A, B] = 0_n$ . Can we conclude that $[A^{-1}, B] = 0_n$ ?

(iii)

Let $A$ , $B$ be $n \times n$ matrices. Suppose that

[A, B] = 0_n, \qquad [A, B]_+ = 0_n

and that $A$ is invertible. Show that $B$ must be the zero matrix.

Problem 7.1.2

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $A$ and $B$ be symmetric $n \times n$ matrices over $\mathbb {R}$ . Show that $AB$ is symmetric if and only if $A$ and $B$ commute.

(ii)

Let $A$ , $B$ be $n \times n$ hermitian matrices. Is $i[A, B]$ hermitian?

(iii)

Let $A$ , $B$ be symmetric $n \times n$ matrices over $\mathbb {R}$ . Show that $[A, B]$ is skew-symmetric over $\mathbb {R}$ .

Problem 7.1.3

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $A$ and $B$ be $n \times n$ matrices over $\mathbb {C}$ . Show that $A$ and $B$ commute if and only if $A - cI_n$ and $B - cI_n$ commute over every $c \in \mathbb {C}$ .

(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ with $[A, B] = 0_n$ . Calculate

[A+cI_n, B+cI_n].

(iii)

Let $\overrightarrow {v}$ be an eigenvector of the $n \times n$ matrix $A$ with eigenvalue $\lambda$ . Show that $\overrightarrow {v}$ is also an eigenvector of $A + cI_n$ , where $c \in \mathbb {C}$ .

Problem 7.1.4

Can one find $2 \times 2$ matrices $A$ and $B$ such that $[A^{2}, B^{2}] = 0_2$ while $[A, B] \neq 0_2$ ?

Origin: Stated according to the original source

Problem 7.1.5

Let $A$ , $B$ , $H$ be $n \times n$ matrices over $\mathbb {C}$ such that $[A, H] = 0_n$ , $[B, H] = 0_n$ . Find $[[A, B], H]$ .

Origin: Stated according to the original source

Problem 7.1.6

(i)(ii)

Let $A$ and $B$ be $n \times n$ hermitian matrices. Suppose that

A^{2} = I_n, \qquad B^{2} = I_n

and

[A, B]_+ \equiv AB + BA = 0_n.

Let $\overrightarrow {x} \in \mathbb {C}^{n}$ be normalized, i.e. $\left\| \overrightarrow {x}\right\| = 1$ . Here $\overrightarrow {x}$ is considered as a column vector.

Origin: Stated according to the original source

(i)

Show that

(\overrightarrow {x}^{*}A\overrightarrow {x})^{2} + (\overrightarrow {x}^{*}B\overrightarrow {x})^{2} \leq 1.

(ii)

Give an example for the matrices $A$ and $B$ .

Problem 7.1.7

Let $A$ , $B$ be skew-hermitian matrices over $\mathbb {C}$ , i.e. $A^{*} = -A$ , $B^{*} = -B$ . Is the commutator of $A$ and $B$ again skew-hermitian?

Origin: Stated according to the original source

Problem 7.1.8

Let $A$ , $B$ be $2 \times 2$ skew-symmetric matrices over $\mathbb {R}$ . Find the commutator $[A, B]$ .

Origin: Stated according to the original source

Problem 7.1.9

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Let $S$ be an invertible $n \times n$ matrix over $\mathbb {C}$ with $\widetilde{A} = S^{-1}AS$ , $\widetilde{B} = S^{-1}BS$ . Show that $[\widetilde{A}, \widetilde{B}] = S^{-1}[A, B]S$ .

(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[A, B] = 0_n$ . Let $U$ be a unitary matrix. Calculate $[U^{*}AU, U^{*}BU]$ .

(iii)

Let $T$ be an invertible matrix. Show that $T^{-1}AT \equiv A + T^{-1}[A, T]$ .

Problem 7.1.10

Can we find $n \times n$ matrices $A$ , $B$ over $\mathbb {C}$ such that

[A, B] = I_n

where $I_n$ denotes the identity matrix?

Origin: Stated according to the original source

Problem 7.1.11

Show that any two $2 \times 2$ matrices which commute with the matrix

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

commute with each other.

Origin: Stated according to the original source

Problem 7.1.12

Can we find $2 \times 2$ matrices $A$ and $B$ of the form

A = \begin{pmatrix} 0 & a_{12} \\ a_{21} & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & b_{12} \\ b_{21} & 0 \end{pmatrix}

and singular (i.e. $\operatorname {det}\left(A\right) = 0$ and $\operatorname {det}\left(B\right) = 0$ ) such that $[A, B]_+ = I_2$ ?

Origin: Stated according to the original source

Problem 7.1.13

Let $A$ be an $n \times n$ hermitian matrix over $\mathbb {C}$ . Assume that the eigenvalues of $A$ , $\lambda_1, \ldots , \lambda_n$ are nondegenerate and that the normalized eigenvectors $\overrightarrow {v}_j$ ( $j=1,\ldots ,n$ ) of $A$ form an orthonormal basis in $\mathbb {C}^{n}$ . Let $B$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $[A, B] = 0_n$ . Show that

\overrightarrow {v}_k^{*}B\overrightarrow {v}_j = 0 \quad \text{for} \quad k \neq j.

Origin: Stated according to the original source

Problem 7.1.14

Let $A$ , $B$ be hermitian $n \times n$ matrices. Assume they have the same set of eigenvectors

A\overrightarrow {v}_j = \lambda _j \overrightarrow {v}_j, \qquad B\overrightarrow {v}_j = \mu _j \overrightarrow {v}_j, \qquad j = 1, \ldots , n

and that the normalized eigenvectors form an orthonormal basis in $\mathbb {C}^{n}$ . Show that $[A, B] = 0_n$ .

Origin: Stated according to the original source

Problem 7.1.15

(i)(ii)

Let $A$ , $B$ be $n \times n$ matrices. Then we have the expansion

e^{A}Be^{-A} = B + [A, B] + \frac{1}{2!}[A, [A, B]] + \frac{1}{3!}[A, [A, [A, B]]] + \cdots .

Origin: Stated according to the original source

(i)

Assume that $[A, B] = A$ . Calculate $e^{A}Be^{-A}$ .

(ii)

Assume that $[A, B] = B$ . Calculate $e^{A}Be^{-A}$ .

Problem 7.1.16

Let $A$ be an arbitrary $n \times n$ matrix over $\mathbb {C}$ with $\operatorname {tr}(A) = 0$ . Show that $A$ can be written as commutator, i.e. there are $n \times n$ matrices $X$ and $Y$ such that $A = [X, Y]$ .

Origin: Stated according to the original source

Problem 7.1.17

Let $A$ , $B$ be $2 \times 2$ symmetric matrices over $\mathbb {R}$ . Assume that $AA^{T} = I_2$ and $BB^{T} = I_2$ . Is $[A, B] = 0_2$ ? Prove or disprove.

Origin: Stated according to the original source

Problem 7.1.18

Let $A$ , $B$ , $X$ , $Y$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $AX - XB = Y$ . Let $z \in \mathbb {C}$ . Show that $(A-zI_n)X - X(B-zI_n) = Y$ .

Origin: Stated according to the original source

Problem 7.1.19

(i)(ii)

Origin: Stated according to the original source

(i)

Can we find nonzero symmetric $2 \times 2$ matrices $H$ and $A$ over $\mathbb {R}$ such that

[H, A] = \mu A

where $\mu \in \mathbb {R}$ and $\mu \neq 0$ ?

(ii)

Let $K$ be a nonzero $n \times n$ hermitian matrix. Let $B$ be a nonzero $n \times n$ matrix. Assume that

[K, B] = aB

where $a \in \mathbb {R}$ and $a \neq 0$ . Show that $B$ cannot be hermitian.

Problem 7.1.20

(i)(ii)

A truncated Bose annihilation operator is defined as the $n \times n$ ( $n \geq 2$ ) matrix

B_n = \begin{pmatrix} 0 & \sqrt{1} & 0 & 0 & 0 & \ldots & 0 \\ 0 & 0 & \sqrt{2} & 0 & 0 & \ldots & 0 \\ 0 & 0 & 0 & \sqrt{3} & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & 0 & \sqrt{n-1} \\ 0 & 0 & 0 & 0 & \ldots & 0 & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Calculate $B_n^{*}B_n$ .

(ii)

Calculate the commutator $[B_n, B_n^{*}]$ .

Problem 7.1.21

Find nonzero $2 \times 2$ matrices $A$ , $B$ such that $[A, B] \neq 0_2$ , but

[A, [A, B]] = 0_2, \qquad [B, [A, B]] = 0_2.

Origin: Stated according to the original source

Problem 7.1.22

Let

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

Find all $2 \times 2$ matrices $A$ such that $[A, C] = 0_2$ .

Origin: Stated according to the original source

Problem 7.1.23

Let $A$ and $B$ be positive semidefinite matrices. Can we conclude that $[A, B]_+$ is positive semidefinite?

Origin: Stated according to the original source

Problem 7.1.24

Let $A$ , $B$ be $n \times n$ matrices. Given the expression

A^{2}B + AB^{2} + B^{2}A + BA^{2} - 2ABA - 2BAB.

Write the expression in a more compact form using commutators.

Origin: Stated according to the original source

Problem 7.1.25

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $A^{2} = I_n$ and $B^{2} = I_n$ . Find the commutators $[AB+BA, A]$ , $[AB+BA, B]$ . Give an example of such matrices for $n=2$ and $A \neq B$ .

(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ such that $A^{2} = I_n$ , $B^{2} = I_n$ . Now assume that $[A, B]_+ = 0_n$ . Show that there is no solution for $A$ and $B$ if $n$ is odd.

Problem 7.1.26

Let $A_1$ , $A_2$ , $A_3$ be $n \times n$ matrices over $\mathbb {C}$ . The ternary commutator $[A_1, A_2, A_3]$ (also called the ternutator) is defined as

[A_1, A_2, A_3] := \sum _{\pi \in S_3} \operatorname {sign}\left(\pi \right) A_{\pi (1)}A_{\pi (2)}A_{\pi (3)} \equiv A_1A_2A_3 + A_2A_3A_1 + A_3A_1A_2 - A_1A_3A_2 - A_2A_1A_3 - A_3A_2A_1.

Let $n = 2$ and consider the Pauli spin matrices $\sigma_1$ , $\sigma_2$ , $\sigma_3$ . Calculate the ternutator $[\sigma_1, \sigma_2, \sigma_3]$ .

Origin: Stated according to the original source

Problem 7.1.27

(i)(ii)

Let $\oplus$ be the direct sum.

Origin: Stated according to the original source

(i)

Let $A$ , $B$ , $H$ be $n \times n$ matrices such that $[H, A] = 0_n$ , $[H, B] = 0_n$ . Show that

[H \oplus I_n + I_n \oplus H, A \oplus B] = 0_{2n}.

(ii)

Let $A_1$ , $A_2$ be $m \times m$ matrices over $\mathbb {C}$ . Let $B_1$ , $B_2$ be $n \times n$ matrices over $\mathbb {C}$ . Show that

[A_1 \oplus B_1, A_2 \oplus B_2] = ([A_1, A_2]) \oplus ([B_1, B_2])

Problem 7.1.28

(i)(ii)

Origin: Stated according to the original source

(i)

Can one find non-invertible $2 \times 2$ matrices $A$ and $B$ such the commutator $[A, B]$ is invertible?

(ii)

Consider the $3 \times 3$ matrices

A = \begin{pmatrix} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & a_3 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & b_1 \\ 0 & b_2 & 0 \\ b_3 & 0 & 0 \end{pmatrix}.

Can we find $a_j$ , $b_j$ ( $j=1,2,3$ ) such that the commutator $[A, B]$ is invertible?

Problem 7.1.29

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $B$ is invertible. Show that

[A, B^{-1}] \equiv -B^{-1}[A, B]B^{-1}.

Origin: Stated according to the original source

Problem 7.1.30

(i)(ii)

Consider the $3 \times 3$ matrices over $\mathbb {C}$

A = \begin{pmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & b_{12} & b_{13} \\ b_{21} & 0 & b_{23} \\ b_{31} & b_{32} & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Calculate the commutator $[A, B]$ and $\operatorname {det}\left({[A, B]}\right)$ .

(ii)

Set $a_{11} = e^{i\phi_1}$ , $a_{22} = e^{i\phi_2}$ , $a_{33} = e^{i\phi_3}$ . Find the condition on $\phi_1$ , $\phi_2$ , $\phi_3$ , $b_{12}$ , $b_{13}$ , $b_{21}$ , $b_{23}$ , $b_{31}$ , $b_{32}$ such that $[A, B]$ is unitary.

Problem 7.1.31

Consider the set of six $3 \times 3$ matrices

A_1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad A_2 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad A_3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},

A_{12} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad A_{23} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad A_{13} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}.

Calculate the anticommutator and thus show that we have a basis of a Jordan algebra.

Origin: Stated according to the original source

Problem 7.1.32

Let $A$ , $B$ be hermitian matrices, i.e. $A^{*} = A$ and $B^{*} = B$ . Then in general $A+iB$ is nonnormal. What are the conditions on $A$ and $B$ such that $A+iB$ is normal?

Origin: Stated according to the original source

Problem 7.1.33

Show that one can find a $3 \times 3$ matrix over $\mathbb {R}$ such that

A^{2}A^{T} + A^{T}A^{2} = 2A, \qquad AA^{T}A = 2A, \qquad A^{3} = 0_3

and

\frac{1}{2}[A^{T}, A] = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 7.1.34

Consider the $3 \times 3$ matrices

D = \begin{pmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{pmatrix}, \qquad M = \begin{pmatrix} 0 & a & 0 \\ a & 0 & b \\ 0 & b & 0 \end{pmatrix}.

Find the conditions on $d_1$ , $d_2$ , $d_3$ , $a$ , $b$ such that

[D, M] = \begin{pmatrix} 0 & a & 0 \\ -a & 0 & b \\ 0 & -b & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 7.1.35

(i)(ii)(iii)(iv)

Origin: Stated according to the original source

(i)

Consider the Pauli spin matrix $\sigma_1$ with eigenvalues $+1$ and $-1$ and corresponding normalized eigenvectors

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}.

Find all $2 \times 2$ matrices $A$ such that $[\sigma_1, A] = 0_2$ . Find the eigenvalues and eigenvectors of $A$ . Discuss.

(ii)

Consider the counter-diagonal unit matrix

J_3 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}

with eigenvalues $+1$ (twice) and $-1$ and the corresponding normalized eigenvectors

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \qquad \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}.

Find all $3 \times 3$ matrices $A$ such that $[J_3, A] = 0_3$ . Find the eigenvalues and normalized eigenvectors of $A$ and compare with the eigenvectors of $J_3$ . Discuss.

(iii)

Consider the $4 \times 4$ matrix (counter-diagonal unit matrix)

J_4 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \sigma _1 \otimes \sigma _1

with the eigenvalues $+1$ (twice) and $-1$ (twice) and the corresponding non-entangled eigenvectors

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \otimes \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix} \otimes \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix},

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \otimes \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}, \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix} \otimes \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}.

Find entangled eigenvectors. Find all $4 \times 4$ matrices $A$ such that $[A, J_4] \equiv AJ_4 - J_4A = 0_4$ .

(iv)

Let $J_n$ be the $n \times n$ counter unit matrix. Let $A$ be an $n \times n$ matrix with $[J_n, A] = 0_n$ . Show that $[J_n \otimes J_n, A \otimes A] = 0_{n^{2}}$ .

Problem 7.1.36

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $M$ , $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that

[M, A] = 0_n, \qquad [M, B] = 0_n.

Calculate the commutator $[M \otimes I_n + I_n \otimes M, A \otimes B]$ .

(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Calculate the commutator $[A \otimes I_n + I_n \otimes A, B \otimes B]$ . Assume that $[A, B] = 0_n$ .

(iii)

Find the commutator

[A \otimes B + B \otimes A, A \otimes A - B \otimes B].

Simplify the result for $[A, B] = 0_n$ . Simplify the result for $A^{2} = B^{2} = 0_n$ . Simplify the result for $A^{2} = B^{2} = I_n$ .

Problem 7.1.37

(i)(ii)

The four Dirac matrices which play a role in the Dirac equation are given by

\gamma _1 = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ i & 0 & 0 & 0 \end{pmatrix}, \qquad \gamma _2 = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix},

\gamma _3 = \begin{pmatrix} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & i \\ i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix}, \qquad \gamma _4 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Show that the matrices can be written as Kronecker product of the Pauli spin matrices $\sigma_1$ , $\sigma_2$ , $\sigma_3$ and the $2 \times 2$ identity matrix $I_2$ .

(ii)

Calculate the anticommutators $[\gamma_4, \gamma_1]_+$ , $[\gamma_4, \gamma_2]_+$ , $[\gamma_4, \gamma_3]_+$ .

Problem 7.1.38

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . What is the condition on $A$ , $B$ such that

[A \otimes B, B \otimes A] = 0_{n^{2}}.

Origin: Stated according to the original source

Problem 7.1.39

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ with commutator $[A, B]$ . Let $R$ be an $n \times n$ matrix over $\mathbb {C}$ with $R^{2} = I_n$ . Find the commutator $[A \otimes R, B \otimes R]$ .

Origin: Stated according to the original source

Problem 7.1.40

(i)(ii)

Let $A_1$ , $A_2$ , $A_3$ be $m \times m$ matrices. Let $B_1$ , $B_2$ , $B_3$ be $n \times n$ matrices. Consider the $(n \cdot m) \times (n \cdot m)$ matrices

X_1 = A_1 \otimes I_n + I_m \otimes B_1, \qquad X_2 = A_2 \otimes I_n + I_m \otimes B_2, \qquad X_3 = A_3 \otimes I_n + I_m \otimes B_3.

Origin: Stated according to the original source

(i)

Calculate the commutators and show that the results can again be expressed using commutators.

(ii)

Assume that

[A_1, A_2] = A_3, \qquad [A_2, A_3] = A_1, \qquad [A_3, A_1] = A_2

and

[B_1, B_2] = B_3, \qquad [B_2, B_3] = B_1, \qquad [B_3, B_1] = B_2.

Use these commutation relations to simplify the results of (i).

Problem 7.1.41

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[A, B] = 0_n$ . Can we conclude that $A \otimes B - B \otimes A = 0_{n^{2}}$ ?

Origin: Stated according to the original source

Problem 7.1.42

Let $A$ be a $2 \times 2$ matrix with $A^{2} = I_2$ . Find the commutator and anticommutator of

X = A \otimes \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = A \otimes \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

Problem 7.1.43

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . We define the quasi-multiplication

A \circ B := \frac{1}{2}(AB+BA).

Obviously $A \circ B = B \circ A$ . Show that $(A^{2} \circ B) \circ A = A^{2} \circ (B \circ A)$ . This is called the Jordan identity.

Origin: Stated according to the original source

Section 7.2Supplementary Problems[36/0%]

Problem 7.2.1

Consider the $2 \times 2$ matrices

A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}, \qquad B(\beta ) = \begin{pmatrix} \cosh (\beta ) & \sinh (\beta ) \\ \sinh (\beta ) & \cosh (\beta ) \end{pmatrix}

where $\alpha , \beta \in \mathbb {R}$ . Calculate the commutator $[A(\alpha ), B(\beta )]$ . What is the condition on $\alpha$ , $\beta$ such that $[A(\alpha ), B(\beta )] = 0_2$ ?

Origin: Stated according to the original source

Problem 7.2.2

Let $A$ be an arbitrary $2 \times 2$ matrix. Show that

[A \otimes \sigma _3, I_2 \otimes A] = A \otimes ([\sigma _3, A]).

Origin: Stated according to the original source

Problem 7.2.3

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Find all nonzero $2 \times 2$ matrices $A$ , $B$ such that $[A, B] = A+B$ . Since $\operatorname {tr}([A, B]) = 0$ we have $\operatorname {tr}(A+B) = 0$ .

(ii)

Find all $2 \times 2$ matrices $A$ and $B$ such that $[A, B] = A-B$ .

(iii)

Find all $2 \times 2$ matrices $A$ and $B$ such that $[A \otimes A, B \otimes B] = A \otimes A - B \otimes B$ .

Problem 7.2.4

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ matrices $A$ over $\mathbb {C}$ such that

[A, A^{*}]_+ = I_2, \qquad [A, A^{*}] = \sigma _3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

From the two conditions we find

AA^{*} + A^{*}A = I_2, \qquad AA^{*} - A^{*}A = I_2.

(ii)

Find all $2 \times 2$ matrices $B$ over $\mathbb {C}$ such that $[B, B^{*}]_+ = I_2$ . Start with

B = \begin{pmatrix} r_{11}e^{i\phi _{11}} & r_{12}e^{i\phi _{12}} \\ r_{21}e^{i\phi _{21}} & r_{22}e^{i\phi _{22}} \end{pmatrix} \quad \Rightarrow \quad B^{*} = \begin{pmatrix} r_{11}e^{-i\phi _{11}} & r_{21}e^{-i\phi _{21}} \\ r_{12}e^{-i\phi _{21}} & r_{22}e^{-i\phi _{22}} \end{pmatrix}.

(iii)

Find all $2 \times 2$ matrices $C$ over $\mathbb {C}$ such that $[C, C^{*}]_+ = I_2$ , $[C, C^{*}] = \sigma_3$ , where $\sigma_3$ is the third Pauli spin matrix.

Problem 7.2.5

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Find all nonzero $2 \times 2$ matrices $K_+$ , $K_-$ , $K_3$ such that

[K_3, K_+] = K_+, \qquad [K_3, K_-] = -K_-, \qquad [K_+, K_-] = -2K_3

where $(K_+)^{*} = K_-$ . Since $\operatorname {tr}([A, B]) = 0$ for any $n \times n$ matrices $A$ , $B$ we have

\operatorname {tr}(K_+) = \operatorname {tr}(K_-) = \operatorname {tr}(K_3) = 0.

(ii)

Find all nonzero $2 \times 2$ matrices $A_1$ , $A_2$ , $A_3$ such that

[A_1, A_2] = 0_2, \qquad [A_1, A_3] = A_1, \qquad [A_2, A_3] = A_2.

(iii)

Find all $2 \times 2$ matrices $A$ , $B$ , $C$ such that $[A, B] \neq 0_2$ , $[A, C] \neq 0_2$ , $[B, C] \neq 0_2$ and

[A, [B, C]] = 0_2.

Problem 7.2.6

Can one find a $2 \times 2$ matrix $A$ over $\mathbb {R}$ such that

[A^{T}, A] = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}?

Origin: Stated according to the original source

Problem 7.2.7

(i)(ii)

Let $\alpha , \beta \in \mathbb {C}$ .

Origin: Stated according to the original source

(i)

Show that the $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} \beta & -\beta ^{2} \\ 1 & -\beta \end{pmatrix}

satisfy the conditions $[A, B]_+ = I_2$ , $[A, A]_+ = 0_2$ , $[B, B]_+ = 0_2$ .

(ii)

Show that the $2 \times 2$ matrices

A = \begin{pmatrix} \alpha & 1 \\ -\alpha ^{2} & -\alpha \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}

satisfy the conditions $[A, B]_+ = I_2$ , $[A, A]_+ = 0_2$ , $[B, B]_+ = 0_2$ .

Problem 7.2.8

(i)(ii)

Origin: Stated according to the original source

(i)

Find all nonzero $2 \times 2$ matrices $A$ , $B$ such that

[A, B]_+ = 0_2, \qquad \operatorname {tr}(AB^{*}) = 0.

(ii)

Find all nonzero $2 \times 2$ matrices $A$ and $B$ such that $[A, [A, B]] = 0_2$ .

Problem 7.2.9

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ , $C$ be $3 \times 3$ nonzero matrices with the commutation relation

[A, B] = C, \qquad [B, C] = A, \qquad [C, A] = B.

Consider the six $27 \times 27$ matrices

A \otimes B \otimes C, \quad A \otimes C \otimes B, \quad B \otimes A \otimes C, \quad B \otimes C \otimes A, \quad C \otimes A \otimes B, \quad C \otimes B \otimes A.

Find all the commutators for these six matrices. Discuss.

(ii)

Let $A_1$ , $A_2$ , $A_3$ be nonzero $3 \times 3$ matrices which satisfy the commutation relations

[A_1, A_2] = A_3, \qquad [A_2, A_3] = A_1, \qquad [A_3, A_1] = A_2.

Let

X := \sum _{k=1}^{3} c_k A_k.

Find the commutator

\left[\sum _{j=1}^{3}(A_j \otimes A_j), X \otimes I_3 + I_3 \otimes X\right].

Problem 7.2.10

(i)(ii)

Origin: Stated according to the original source

(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Show that if $A$ and $B$ commute and if $A$ is normal, then $A^{*}$ and $B$ commute.

(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . It can be written as $A = HU$ , where $H$ is a non-negative definite hermitian matrix and $U$ is unitary. Show that the matrices $H$ and $A$ commute if and only if $A$ is normal.

Problem 7.2.11

(i)(ii)(iii)(iv)

Let $\sigma_j$ ( $j=0,1,2,3$ ) be the Pauli spin matrices, where $\sigma_0 = I_2$ . Form the four $4 \times 4$ Dirac matrices

\gamma _k = \begin{pmatrix} 0_2 & \sigma _k \\ -\sigma _k & 0_2 \end{pmatrix}, \qquad k = 0,1,2,3.

Origin: Stated according to the original source

(i)

Are the matrices $\gamma_k$ linearly independent?

(ii)

Find the eigenvalues and eigenvectors of the $\gamma_k$ 's.

(iii)

Are the matrices $\gamma_k$ invertible. Use the result from (ii). If so, find the inverse.

(iv)

Find the commutators $[\gamma_k, \gamma_\ell ]$ for $k, \ell = 0,1,2,3$ . Find the anticommutators $[\gamma_k, \gamma_\ell ]_+$ for $k, \ell = 0,1,2,3$ .

Problem 7.2.12

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[A, B] \neq 0_n$ . Can we conclude that

[e^{A}, e^{B}] \neq 0_n?

Origin: Stated according to the original source

Problem 7.2.13

Let $A$ , $B$ be invertible $n \times n$ matrices over $\mathbb {C}$ . Assume that $[A, B] = 0_n$ . Can we conclude that $[A^{-1}, B^{-1}] = 0_n$ ?

Origin: Stated according to the original source

Problem 7.2.14

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\sigma_1$ , $\sigma_2$ , $\sigma_3$ be the Pauli spin matrices. Consider the three nonnormal matrices

A = \sigma _1+i\sigma _2, \qquad B = \sigma _2+i\sigma _3, \qquad C = \sigma _3+i\sigma _1.

Find the commutators and anticommutators. Discuss.

(ii)

Consider the three nonnormal matrices

X = \sigma _1 \otimes \sigma _1+i\sigma _2 \otimes \sigma _2, \qquad Y = \sigma _2 \otimes \sigma _2+i\sigma _3 \otimes \sigma _3, \qquad Z = \sigma _3 \otimes \sigma _3+i\sigma _1 \otimes \sigma _1.

Find the commutators and anticommutators. Discuss.

Problem 7.2.15

(i)(ii)(iii)(iv)(v)

Origin: Stated according to the original source

(i)

Let $\sigma_1$ , $\sigma_2$ , $\sigma_3$ be the Pauli spin matrices. Consider the nonnormal matrices

A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \qquad B = \frac{1}{2}\begin{pmatrix} 3 & -1 \\ 1 & 1 \end{pmatrix}, \qquad C = \frac{1}{2}\begin{pmatrix} 3 & i \\ i & 1 \end{pmatrix}

where $\operatorname {det}\left(A\right) = \operatorname {det}\left(B\right) = \operatorname {det}\left(C\right) = 1$ , i.e. $A$ , $B$ , $C$ are elements of the Lie group $SL(2, \mathbb {C})$ . Show that $[A, A^{*}] = \sigma_3$ , $[B, B^{*}] = \sigma_1$ , $[C, C^{*}] = \sigma_2$ .

(ii)

Consider the unitary matrices

U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad V = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}.

Show that $B = UAU^{*}$ , $C = VBV^{*}$ .

(iii)

Consider the nonnormal and noninvertible matrices

X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = \frac{1}{2}\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}, \qquad Z = \frac{1}{2}\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}.

All have trace zero and thus are elements of the Lie algebra $\mathfrak {sl}(2, \mathbb {C})$ . Show that

[X, X^{*}] = \sigma _3, \qquad [Y, Y^{*}] = \sigma _1, \qquad [Z, Z^{*}] = \sigma _2.

Hint. Obviously we have $X = A - I_2$ , $Y = B - I_2$ , $Z = C - I_2$ .

(iv)

Show that $Y = UXU^{*}$ , $Z = VYV^{*}$ .

(v)

Study the commutators

[X \otimes X, X^{*} \otimes X^{*}], \qquad [X \otimes X^{*}, X^{*} \otimes X],

[Y \otimes Y, Y^{*} \otimes Y^{*}], \qquad [Y \otimes Y^{*}, Y^{*} \otimes Y],

[Z \otimes Z, Z^{*} \otimes Z^{*}], \qquad [Z \otimes Z^{*}, Z^{*} \otimes Z].

Problem 7.2.16

(i)(ii)

Origin: Stated according to the original source

(i)

Can one find $n \times n$ matrices $A$ and $B$ over $\mathbb {C}$ such that the following conditions are satisfied $[A, B] = 0_n$ , $[A, B]_+ = 0_n$ and both $A$ and $B$ are invertible?

(ii)

Can one find $n \times n$ matrices $A$ and $B$ over $\mathbb {C}$ such that the following conditions are satisfied

[A, B] = 0_n, \qquad [A, B]_+ = I_n

and both $A$ and $B$ are invertible?

Problem 7.2.17

Consider the hermitian $3 \times 3$ matrices

L_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \qquad L_2 = \begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \end{pmatrix}, \qquad L_3 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.

Find the commutators $[L_1, L_2]$ , $[L_2, L_3]$ , $[L_3, L_1]$ . Find the $3 \times 3$ matrix

A = c_{23}[L_2, L_3]_+ + c_{13}[L_3, L_1]_+ + c_{12}[L_1, L_2]_+.

Origin: Stated according to the original source

Problem 7.2.18

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Find $4 \times 4$ matrices $C$ and $2 \times 2$ matrices $A$ such that

[C, A \otimes I_2 + I_2 \otimes A] = 0_4.

(ii)

Find the conditions on the two $2 \times 2$ hermitian matrices $B$ , $C$ such that

[B \otimes C, P] = 0_4

where $P$ is the permutation matrix

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

(iii)

Let $E$ , $F$ be arbitrary $2 \times 2$ matrices. Then

[E \otimes I_2, I_2 \otimes F] = 0_4.

Let $X$ , $Y$ be $4 \times 4$ matrices with $[X, Y] = 0_4$ . Can $X$ , $Y$ be written as $X = E \otimes I_2$ , $Y = I_2 \otimes F$ or $Y = E \otimes I_2$ , $X = I_2 \otimes F$ ?

Problem 7.2.19

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Consider the two $2 \times 2$ matrices (counter diagonal matrices)

A = \begin{pmatrix} 0 & a_{12} \\ a_{21} & 0 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & b_{12} \\ b_{21} & 0 \end{pmatrix}.

Find the condition on $A$ and $B$ such that $[A, B] = 0_2$ .

(ii)

Consider the two $3 \times 3$ matrices (counter diagonal matrices)

A = \begin{pmatrix} 0 & 0 & a_{13} \\ 0 & a_{22} & 0 \\ a_{31} & 0 & 0 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & 0 & b_{13} \\ 0 & b_{22} & 0 \\ b_{31} & 0 & 0 \end{pmatrix}.

Find the condition on $A$ and $B$ such that $[A, B] = 0_3$ .

(iii)

Extend to $n$ dimensions.

Problem 7.2.20

Consider $(m+n) \times (m+n)$ matrices of the form

\begin{pmatrix} m \times m & m \times n \\ n \times m & n \times n \end{pmatrix}.

Let

B = \begin{pmatrix} B_1 & 0 \\ 0 & B_2 \end{pmatrix}, \qquad \widetilde{B} = \begin{pmatrix} \widetilde{B}_1 & 0 \\ 0 & \widetilde{B}_2 \end{pmatrix}, \qquad F = \begin{pmatrix} 0 & F_1 \\ F_2 & 0 \end{pmatrix}, \qquad \widetilde{F} = \begin{pmatrix} 0 & \widetilde{F}_1 \\ \widetilde{F}_2 & 0 \end{pmatrix}.

Show that

[B, \widetilde{B}] = \begin{pmatrix} B_1\widetilde{B}_1 - \widetilde{B}_1B_1 & 0 \\ 0 & B_2\widetilde{B}_2 - \widetilde{B}_2B_2 \end{pmatrix},

[B, F] = \begin{pmatrix} 0 & B_1F_1 - F_1B_2 \\ B_2F_2 - F_2B_1 & 0 \end{pmatrix},

[F, \widetilde{F}]_+ = \begin{pmatrix} F_1\widetilde{F}_2 + \widetilde{F}_1F_2 & 0 \\ 0 & F_2\widetilde{F}_1 + \widetilde{F}_2F_1 \end{pmatrix}.

Origin: Stated according to the original source

Problem 7.2.21

Let $n \geq 1$ and $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Show that

[A, B^{n}] = \sum _{j=0}^{n-1} A^{j}[A, B]B^{n-1-j}.

Origin: Stated according to the original source

Problem 7.2.22

Let $A_1$ , $A_2$ , $B_1$ , $B_2$ be $n \times n$ matrices over $\mathbb {C}$ with $A_1^{2} = A_2^{2} = B_1^{2} = B_2^{2} = I_n$ . Let

T = A_1 \otimes (B_1+B_2) + A_2 \otimes (B_1-B_2).

Show that $T^{2} = 4I_n \otimes I_n - [A_1, A_2] \otimes [B_1, B_2]$ .

Origin: Stated according to the original source

Problem 7.2.23

(i)(ii)

Let $A$ , $B$ be $2 \times 2$ matrices.

Origin: Stated according to the original source

(i)

Solve the equation $A \otimes B - B \otimes A = [A, B] \otimes I_2 + I_2 \otimes [A, B]$ .

(ii)

Solve the equation $A \otimes B - B \otimes A = [A \otimes A, B \otimes B]$ .

Problem 7.2.24

Let $A$ be an $n \times n$ matrix. Find the commutator

\left[ \begin{pmatrix} 0_n & A \\ A & 0_n \end{pmatrix}, \begin{pmatrix} 0_n & A \\ -A & 0_n \end{pmatrix} \right]

and the anticommutator.

Origin: Stated according to the original source

Problem 7.2.25

Let $A$ , $B$ be $n \times n$ matrices and $T$ a (fixed) invertible $n \times n$ matrix. We define the bracket

[A, B]_T := ATB - BTA.

Let

T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Find $[X, Y]_T$ , $[X, H]_T$ , $[Y, H]_T$ .

Origin: Stated according to the original source

Problem 7.2.26

Can we find $3 \times 3$ matrices $A$ and $B$ such that $[A, B]_+ = 0_3$ and $A^{2} = B^{2} = I_3$ ?

Origin: Stated according to the original source

Problem 7.2.27

(i)(ii)

Origin: Stated according to the original source

(i)

Let $X$ , $Y$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[X, Y] = 0_n$ . Do $X \otimes X$ and $Y \otimes Y$ commute? Do $X \otimes X$ and $Y \otimes I_n + I_n \otimes X$ commute?

(ii)

Let $X$ , $Y$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[X, Y]_+ = 0_n$ . Do $X \otimes X$ and $Y \otimes Y$ anticommute? Do $X \otimes X$ and $Y \otimes I_n + I_n \otimes X$ anticommute?

Problem 7.2.28

Find all $2 \times 2$ matrices over $\mathbb {C}$ such that the commutator is an invertible diagonal matrix $D$ , i.e. $d_{11} \neq 0$ and $d_{22} \neq 0$ . An example is

A = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \quad \Rightarrow \quad D = \begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}.

Origin: Stated according to the original source

Problem 7.2.29

Consider the skew-symmetric $3 \times 3$ matrices

A_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad A_2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \qquad A_3 = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

with $[A_1, A_2] = A_3$ , $[A_2, A_3] = A_1$ , $[A_3, A_1] = A_2$ . Find the commutators for the $9 \times 9$ matrices

A_1 \otimes I_3 + I_3 \otimes A_1, \qquad A_2 \otimes I_3 + I_3 \otimes A_2, \qquad A_3 \otimes I_3 + I_3 \otimes A_3.

Origin: Stated according to the original source

Problem 7.2.30

Let $A$ , $B$ , $C$ be $n \times n$ matrices. Show that

\operatorname {tr}([A, B]C) \equiv \operatorname {tr}(A[B, C])

applying cyclic invariance.

Origin: Stated according to the original source

Problem 7.2.31

Show that $A = i\sigma_1/2$ , $B = -i\sigma_2$ satisfy

-A = [B, [B, A]], \qquad -B = [A, [A, B]].

Origin: Stated according to the original source

Problem 7.2.32

Consider the linear operators $J_+$ , $J_-$ , $J_z$ satisfying the commutation relations $[J_z, J_+] = J_+$ , $[J_z, J_-] = -J_-$ , $[J_+, J_-] = 2J_z$ , where $(J_+)^{*} = J_-$ . Show that the $2 \times 2$ matrices

J_+ = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad J_- = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad J_z = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

satisfy these commutations relations.

Origin: Stated according to the original source

Problem 7.2.33

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the $3 \times 3$ matrices

P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{13} & b_{11} & b_{12} \\ b_{12} & b_{13} & b_{11} \end{pmatrix}.

Both matrices are circulant matrices. Find the commutator $[P, B]$ . Discuss.

(ii)

Consider the invertible $3 \times 3$ matrices

P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \qquad A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{2i\pi /3} & 0 \\ 0 & 0 & e^{-2i\pi /3} \end{pmatrix}.

Is the matrix $[A, P]$ invertible?

Problem 7.2.34

Let $c \in \mathbb {R}$ and $A$ be an $2 \times 2$ matrix over $\mathbb {R}$ . Find the commutator of the $3 \times 3$ matrices $c \oplus A$ and $A \oplus c$ , where $\oplus$ denotes the direct sum.

Origin: Stated according to the original source

Problem 7.2.35

Let $U$ , $V$ be $n \times n$ unitary matrices. Show that

[V^{-1} \otimes U, V \otimes U^{-1}] = 0_n.

Origin: Stated according to the original source

Problem 7.2.36

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . We define (Jordan product)

A \circ B = \frac{1}{2}(AB+BA).

Show that $A \circ B$ is commutative and satisfies $A \circ (A^{2} \circ B) = A^{2} \circ (A \circ B)$ , but is not associative in general, i.e. $(A \circ B) \circ C \neq A \circ (B \circ C)$ in general.

Origin: Stated according to the original source