Math Problems Vault

Problem 7.1.1

(i)(ii)(iii)

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(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)

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(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)

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(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 $[A, B] \neq 0_2$ ?

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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 $[[A, B], H]$ .

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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.

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(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?

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Problem 7.1.8

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

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Problem 7.1.9

(i)(ii)(iii)

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(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?

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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.

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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 $[A, B]_+ = I_2$ ?

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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.

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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 $[A, B] = 0_n$ .

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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 .

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(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 $A = [X, Y]$ .

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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.

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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 $(A-zI_n)X - X(B-zI_n) = Y$ .

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Problem 7.1.19

(i)(ii)

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(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}.

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(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.

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Problem 7.1.22

Let

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

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

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Problem 7.1.23

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

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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.

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Problem 7.1.25

(i)(ii)

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(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 $[\sigma_1, \sigma_2, \sigma_3]$ .

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Problem 7.1.27

(i)(ii)

Let $\oplus$ be the direct sum.

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(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)

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(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}.

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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}.

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(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.

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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?

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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}.

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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}.

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Problem 7.1.35

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

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(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)

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(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}.

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(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}}.

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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 $[A \otimes R, B \otimes R]$ .

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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.

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(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 $A \otimes B - B \otimes A = 0_{n^{2}}$ ?

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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}.

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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.

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