All textbooks/Problems and Solutions in Introductory and Advanced Matrix Calculus

Chapter 18

Groups, Lie Groups and Matrices

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18.1 · Chapter Problems

18.2 · Supplementary Problems

Section 18.1Chapter Problems[45/100%]

Problem 18.1.1

Let $\phi \in \mathbb {R}$ . Do the $2 \times 2$ matrices

I_{2} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad U(\phi ) = \begin{pmatrix} 0 & e^{i\phi } \\ e^{-i\phi } & 0 \end{pmatrix}

form a group under matrix multiplication?

Origin: Stated according to the original source

Problem 18.1.2

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

Origin: Stated according to the original source

(i)

Show that the $2 \times 2$ matrix

U = \begin{pmatrix} 0 & 1 \\ i & 0 \end{pmatrix}

is unitary.

(ii)

Find the eigenvalues and normalized eigenvectors of $U$ .

(iii)

Find the group generated by $U$ .

(iv)

Find the group generated by $U \otimes U$ .

Problem 18.1.3

Find the group generated by the $3 \times 3$ matrices

G_{1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \qquad G_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.

Set $G_{0} = G_{1}^{2} = I_{3}$ .

Origin: Stated according to the original source

Problem 18.1.4

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the permutation group $S_{3}$ which consists of 6 elements. The matrix representation of the permutations $(12)$ and $(13)$ are

(12) \mapsto \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad (13) \mapsto \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}.

Can the other permutations of $S_{3}$ be constructed from matrix products of these two matrices?

(ii)

Consider the two $3 \times 3$ permutation matrices

C_{1} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \qquad A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}.

Can the remaining four $3 \times 3$ matrices be generated from $C_{1}$ and $A$ using matrix multiplication?

Problem 18.1.5

Find the group generated by the two $2 \times 2$ matrices

\sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad R = \frac{1}{2} \begin{pmatrix} -1 & -\sqrt{3} \\ \sqrt{3} & -1 \end{pmatrix}

under matrix multiplication. Is the group commutative?

Origin: Stated according to the original source

Problem 18.1.6

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

Origin: Stated according to the original source

(i)

Show that the set of matrices

E = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad C_{3} = \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & -1/2 \end{pmatrix}, \quad C_{3}^{-1} = \begin{pmatrix} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{pmatrix},

\sigma _{1} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \sigma _{2} = \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2 & 1/2 \end{pmatrix}, \quad \sigma _{3} = \begin{pmatrix} -1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{pmatrix}

form a group $G$ under matrix multiplication, where $C_{3}^{-1}$ is the inverse matrix of $C_{3}$ .

(ii)

Find the determinant of all these matrices. Does the set of numbers

\left\{ \operatorname {det}\left(E\right), \ \operatorname {det}\left(C_{3}\right), \ \operatorname {det}\left(C_{3}^{-1}\right), \ \operatorname {det}\left(\sigma _{1}\right), \ \operatorname {det}\left(\sigma _{2}\right), \ \operatorname {det}\left(\sigma _{3}\right)\right\}

form a group under multiplication?

(iii)

Find two proper subgroups.

(iv)

Find the right coset decomposition. Find the left coset decomposition.

We obtain the right coset decomposition as follows: Let $G$ be a finite group of order $g$ having a proper subgroup $\mathcal{H}$ of order $h$ . Take some element $g_{2}$ of $G$ which does not belong to the subgroup $\mathcal{H}$ , and make a right coset $\mathcal{H}g_{2}$ . If $\mathcal{H}$ and $\mathcal{H}g_{2}$ do not exhaust the group $G$ , take some element $g_{3}$ of $G$ which is not an element of $\mathcal{H}$ and $\mathcal{H}g_{2}$ , and make a right coset $\mathcal{H}g_{3}$ . Continue making right cosets $\mathcal{H}g_{j}$ in this way. If $G$ is a finite group, all elements of $G$ will be exhausted in a finite number of steps and we obtain the right coset decomposition.

Problem 18.1.7

We know that the set of $2 \times 2$ matrices

E = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad C_{3} = \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & -1/2 \end{pmatrix}, \quad C_{3}^{-1} = \begin{pmatrix} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{pmatrix}

\sigma _{1} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \sigma _{2} = \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2 & 1/2 \end{pmatrix}, \quad \sigma _{3} = \begin{pmatrix} -1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{pmatrix}

forms a group $G$ under matrix multiplication, where $C_{3}^{-1}$ is the inverse matrix of $C_{3}$ . The set of matrices ( $3 \times 3$ permutation matrices)

I = P_{0} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad P_{1} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix},

P_{2} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \qquad P_{3} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix},

P_{4} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \qquad P_{5} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}

also forms a group $G$ under matrix multiplication. Are the two groups isomorphic? A homomorphism which is $1-1$ and onto is an isomorphism.

Origin: Stated according to the original source

Problem 18.1.8

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Show that the $3 \times 3$ matrices

A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix},

C = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \qquad D = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}

form a group under matrix multiplication.

(ii)

Show that the $4 \times 4$ matrices

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

V = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, \qquad W = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}

form a group under matrix multiplication.

(iii)

Show that the two groups (so-called Vierergruppe) are isomorphic.

Problem 18.1.9

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $x \in \mathbb {R}$ . Show that the $2 \times 2$ matrices

A(x) = \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}

form a group under matrix multiplication.

(ii)

Is the group commutative?

(iii)

Find a group that is isomorphic to this group.

Problem 18.1.10

Let $a, b, c, d \in \mathbb {Z}$ . Show that the $2 \times 2$ matrices

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

with $ad - bc = 1$ form a group under matrix multiplication.

Origin: Stated according to the original source

Problem 18.1.11

(i)(ii)

Origin: Stated according to the original source

(i)

Let $c \in \mathbb {R}$ and $c \neq 0$ . Do the $4 \times 4$ matrices

A(c) = \begin{pmatrix} c & c & c & c \\ c & c & c & c \\ c & c & c & c \\ c & c & c & c \end{pmatrix}

form a group under matrix multiplication?

(ii)

Find the eigenvalues of $A(c)$ .

Problem 18.1.12

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

The Heisenberg group is the set of upper $3 \times 3$ matrices of the form

H = \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}

where $a, b, c$ can be taken from some (arbitrary) commutative ring.

Origin: Stated according to the original source

(i)

Find the inverse of $H$ .

(ii)

Given two elements $x, y$ of a group $G$ , we define the commutator of $x$ and $y$ , denoted by $[x, y]$ to be the element $x^{-1}y^{-1}xy$ . If $a, b, c$ are integers (in the ring $\mathbb {Z}$ of the integers) we obtain the discrete Heisenberg group $H_{3}$ . It has two generators

x = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}.

Find $z = xyx^{-1}y^{-1}$ . Show that $xz = zx$ and $yz = zy$ , i.e. $z$ is the generator of the center of $H_{3}$ .

(iii)

The derived subgroup (or commutator subgroup) of a group $G$ is the subgroup $[G, G]$ generated by the set of commutators of every pair of elements of $G$ . Find $[G, G]$ for the Heisenberg group.

(iv)

Let

A = \begin{pmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{pmatrix}

and $a, b, c \in \mathbb {R}$ . Find $\mathrm{exp}\left(A\right)$ .

(v)

The Heisenberg group is a simple connected Lie group whose Lie algebra consists of matrices

L = \begin{pmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{pmatrix}.

Find the commutators $[L, L']$ and $[[L, L'], L']$ , where $[L, L'] := LL' - L'L$ .

Problem 18.1.13

Define

M : \mathbb {R}^{3} \rightarrow V := \left\{ \overrightarrow {a} \cdot \overrightarrow {\sigma } : \overrightarrow {a} \in \mathbb {R}^{3}\right\} \subset \left\{ 2 \times 2 \text{ complex matrices}\right\}

\overrightarrow {a} \rightarrow M(\overrightarrow {a}) = \overrightarrow {a} \cdot \overrightarrow {\sigma } = a_{1}\sigma _{1} + a_{2}\sigma _{2} + a_{3}\sigma _{3}.

This is a linear bijection between $\mathbb {R}^{3}$ and $V$ . Each $U \in SU(2)$ determines a linear map $S(U)$ on $\mathbb {R}^{3}$ by

M(S(U)\overrightarrow {a}) = U^{-1}M(\overrightarrow {a})U.

The right-hand side is clearly linear in $\overrightarrow {a}$ . Show that $U^{-1}M(\overrightarrow {a})U$ is in $V$ , that is, of the form $M(\overrightarrow {b})$ .

Origin: Stated according to the original source

Problem 18.1.14

Show that the nonnormal $2 \times 2$ matrices

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

are conjugate in $SL(2, \mathbb {C})$ but not in $SL(2, \mathbb {R})$ (the real matrices in $SL(2, \mathbb {C})$ ).

Origin: Stated according to the original source

Problem 18.1.15

(i)(ii)

Origin: Stated according to the original source

(i)

Let $G$ be a finite set of real $n \times n$ matrices $\left\{ A_{j}\right\}$ , $1 \leq j \leq r$ , which forms a group under matrix multiplication. Suppose that

\operatorname {tr}\left(\sum _{j=1}^{r} A_{j}\right) \equiv \sum _{j=1}^{r} \operatorname {tr}(A_{j}) = 0

where $\operatorname {tr}$ denotes the trace. Show that

\sum _{j=1}^{r} A_{j} = 0_{n}.

(ii)

Let $\omega := \mathrm{exp}\left(2\pi i/3\right)$ . Show that the $2 \times 2$ matrices

B_{1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B_{2} = \begin{pmatrix} \omega & 0 \\ 0 & \omega ^{2} \end{pmatrix}, \quad B_{3} = \begin{pmatrix} \omega ^{2} & 0 \\ 0 & \omega \end{pmatrix}

B_{4} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad B_{5} = \begin{pmatrix} 0 & \omega \\ \omega ^{2} & 0 \end{pmatrix}, \quad B_{6} = \begin{pmatrix} 0 & \omega ^{2} \\ \omega & 0 \end{pmatrix}

form a group under matrix multiplication. Show that

\sum _{j=1}^{6} \operatorname {tr}(B_{j}) = 0.

Problem 18.1.16

(i)(ii)

Consider the unitary $2 \times 2$ matrix

J := \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Find all $2 \times 2$ matrices $A$ over $\mathbb {R}$ such that $A^{T}JA = J$ .

(ii)

Do these $2 \times 2$ matrices form a group under matrix multiplication?

Problem 18.1.17

Let $J$ be the $2n \times 2n$ matrix

J := \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix}.

Show that the $2n \times 2n$ matrices $A$ satisfying $A^{T}JA = J$ form a group under matrix multiplication. This group is called the symplectic group $Sp(2n)$ .

Origin: Stated according to the original source

Problem 18.1.18

The Pauli spin matrix $\sigma_{1}$ is not only hermitian, unitary and its own inverse, but also a permutation matrix. Find all $2 \times 2$ matrices $A$ such that $\sigma_{1}^{-1}A\sigma_{1} = A$ .

Origin: Stated according to the original source

Problem 18.1.19

For the vector space of the $n \times n$ matrices over $\mathbb {R}$ we can introduce a scalar product via

\left\langle A, B \right\rangle := \operatorname {tr}(AB^{T}).

Consider the Lie group $SL(2, \mathbb {R})$ of the $2 \times 2$ matrices with determinant 1. Find $X, Y \in SL(2, \mathbb {R})$ such that

\left\langle X, Y \right\rangle = 0

where neither $X$ nor $Y$ can be $2 \times 2$ identity matrix.

Origin: Stated according to the original source

Problem 18.1.20

The numbers $1, i, -1, -i$ form a group under multiplication. Do the vectors

\overrightarrow {v}_{1} = \frac{1}{2} \begin{pmatrix} 1 \\ i \\ -1 \\ -i \end{pmatrix}, \quad \overrightarrow {v}_{2} = \frac{1}{2} \begin{pmatrix} i \\ -1 \\ -i \\ 1 \end{pmatrix}, \quad \overrightarrow {v}_{3} = \frac{1}{2} \begin{pmatrix} -1 \\ -i \\ 1 \\ i \end{pmatrix}, \quad \overrightarrow {v}_{4} = \frac{1}{2} \begin{pmatrix} -i \\ 1 \\ i \\ -1 \end{pmatrix}

form an orthonormal basis in $\mathbb {C}^{4}$ ?

Origin: Stated according to the original source

Problem 18.1.21

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the group $G$ of all $4 \times 4$ permutation matrices. Show that

\frac{1}{\left|G\right|} \sum _{g \in G} g

is a projection matrix. Here $\left|G\right|$ denotes the number of elements in the group.

(ii)

Consider the subgroup given by the matrices

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

Show that

\frac{1}{\left|G\right|} \sum _{g \in G} g

is a projection matrix.

Problem 18.1.22

Let $A$ be the $2 \times 2$ matrix

A = \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}

where $r_{jk} \in \mathbb {R}$ , $r_{jk} > 0$ for $j, k = 1, 2$ and $r_{12} = r_{21}$ . We also have $\phi_{jk} \in \mathbb {R}$ for $j, k = 1, 2$ and impose $\phi_{12} = \phi_{21}$ . What are the conditions on $r_{jk}$ and $\phi_{jk}$ such that $I_{2} + iA$ is a unitary matrix?

Origin: Stated according to the original source

Problem 18.1.23

Show that the four $2 \times 2$ matrices

A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad D = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}

from a group under matrix multiplication. Is the group abelian?

Origin: Stated according to the original source

Problem 18.1.24

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Consider the symmetric $2 \times 2$ matrix

A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{11} \end{pmatrix}, \qquad a_{11}, a_{12} \in \mathbb {R}.

Find all invertible $2 \times 2$ matrices $S$ over $\mathbb {R}$ such that $SAS^{-1} = A$ . Obviously the identity matrix $I_{2}$ would be such as matrix.

(ii)

Do the matrices $S$ form a group under matrix multiplication? Prove or disprove.

(iii)

Use the result form (i) to calculate $(S \otimes S)(A \otimes A)(S \otimes S)^{-1}$ . Discuss.

Problem 18.1.25

Find all $2 \times 2$ matrices $S$ over $\mathbb {C}$ with determinant 1 (i.e. they are elements of $SL(2, \mathbb {C})$ ) such that

S^{-1} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} S = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Obviously, the $2 \times 2$ identity matrix is such an element.

Origin: Stated according to the original source

Problem 18.1.26

(i)(ii)

Origin: Stated according to the original source

(i)

Show that the $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}

are similar. This means find an invertible $2 \times 2$ matrix $S$ , i.e. $S \in GL(2, \mathbb {R})$ , such that $SAS^{-1} = B$ .

(ii)

Is there an invertible $2 \times 2$ matrix $S$ such that

(S \otimes S)(A \otimes A)(S^{-1} \otimes S^{-1}) = B \otimes B?

Problem 18.1.27

Find all $M \in GL(2, \mathbb {C})$ such that

M^{-1} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} M = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

Thus we consider the invariance of the Pauli spin matrix $\sigma_{2}$ . Show that these matrices form a group under matrix multiplication.

Origin: Stated according to the original source

Problem 18.1.28

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

Consider the cyclic $3 \times 3$ matrix

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

Origin: Stated according to the original source

(i)

Show that the matrices $C, C^{2}, C^{3}$ form a group under matrix multiplication. Is the group commutative?

(ii)

Find the eigenvalues of $C$ and show that the form a group under multiplication.

(iii)

Find the normalized eigenvalues of $C$ . Show that the form a orthonormal basis in $\mathbb {C}^{3}$ .

(iv)

Use the eigenvalues and normalized eigenvectors to write down the spectral decomposition of $C$ .

Problem 18.1.29

(i)(ii)

Origin: Stated according to the original source

(i)

Find the group generated by the $2 \times 2$ matrix

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

under matrix multiplication. Is the group commutative? Find the determinant of all these matrices. Do these (real) numbers form a group under multiplication?

(ii)

Find the group generated by the $4 \times 4$ matrix

A \otimes A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

under matrix multiplication.

Problem 18.1.30

Let $n \geq 2$ . An invertible integer matrix, $A \in GL_{n}(\mathbb {Z})$ , generates a toral automorphism $f : \mathbb {T}^{n} \rightarrow \mathbb {T}^{n}$ via the formula

f \circ \pi = \pi \circ A, \qquad \pi : \mathbb {R}^{n} \rightarrow \mathbb {T}^{n} := \mathbb {R}^{n}/\mathbb {Z}^{n}.

The set of fixed points of $f$ is given by

\# \text{Fix}(f) := \left\{ x^{*} \in \mathbb {T}^{n} : f(x^{*}) = x^{*}\right\} .

Now we have: if $\operatorname {det}\left(I_{n}-A\right) \neq 0$ , then

\# \text{Fix}(f) = \left|\operatorname {det}\left(I_{n}-A\right)\right|.

Let $n = 2$ and

A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}.

Show that $\operatorname {det}\left(I_{2}-A\right) \neq 0$ and find $\# \text{Fix}(f)$ .

Origin: Stated according to the original source

Problem 18.1.31

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Consider the $2n \times 2n$ matrices

S = \begin{pmatrix} I_{n} & I_{n} \\ A & I_{n}+A \end{pmatrix}, \qquad \widetilde{S} = \begin{pmatrix} A+2I_{n} & -I_{n} \\ I_{n} & 0_{n} \end{pmatrix}.

Can we find an invertible $2n \times 2n$ matrix $T$ such that $\widetilde{S} = T^{-1}ST$ ?

Origin: Stated according to the original source

Problem 18.1.32

Let $R \in \mathbb {C}^{m \times m}$ and $S \in \mathbb {C}^{n \times n}$ be nontrivial involutions. This means that $R = R^{-1} \neq \pm I_{m}$ and $S = S^{-1} \neq I_{n}$ . A matrix $A \in \mathbb {C}^{m \times n}$ is called $(R, S)$ -symmetric if $RAS = A$ . Consider the case $m = n = 2$ and the Pauli spin matrices

R = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad S = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

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

Origin: Stated according to the original source

Problem 18.1.33

Find all $2 \times 2$ matrices $A$ over $\mathbb {R}$ such that $\operatorname {det}\left(A\right) = 1$ (i.e. $A$ is an element of the Lie group $SL(2, \mathbb {R})$ ) and

A \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}.

Do these matrices form a group under matrix multiplication?

Origin: Stated according to the original source

Problem 18.1.34

Let $\alpha \in \mathbb {R}$ . Consider the hermitian matrix which is an element of the noncompact Lie group $SO(1,1)$

A(\alpha ) = \begin{pmatrix} \cosh (\alpha ) & \sinh (\alpha ) \\ \sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix}.

Find the Cayley transform

B = (A-iI_{2})(A+iI_{2})^{-1}.

Note that $B$ is a unitary matrix and therefore an element of the compact Lie group $U(n)$ . Find $B(\alpha \rightarrow \infty )$ .

Origin: Stated according to the original source

Problem 18.1.35

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the Lie group $SL(n, \mathbb {C})$ , i.e. the $n \times n$ matrices over $\mathbb {C}$ with determinant 1. Can we find $A, B \in SL(n, \mathbb {C})$ such that $[A, B]$ is an element of $SL(n, \mathbb {C})$ ?

(ii)

Consider the $2 \times 2$ matrices

A(z) = \begin{pmatrix} 1 & 0 \\ z & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.

Both are elements of the non-compact Lie group $SL(2, \mathbb {C})$ . Can one finds $z \in \mathbb {C}$ such that the commutator $[A(z), B]$ is again an element of $SL(2, \mathbb {C})$ ?

Problem 18.1.36

The Lie group $SU(2)$ is defined by

SU(2) := \left\{ U \ 2 \times 2 \text{ matrix} : UU^{*} = I_{2}, \ \operatorname {det}\left(U\right) = 1\right\} .

Let (3-sphere)

S^{3} := \left\{ (x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb {R}^{4} : x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2} = 1\right\} .

Show that $SU(2)$ can be identified as a real manifold with the 3-sphere $S^{3}$ .

Origin: Stated according to the original source

Problem 18.1.37

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

Consider the matrix ( $\alpha \in \mathbb {R}$ )

A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) \\ \sin (\alpha ) & -\cos (\alpha ) \end{pmatrix}.

Origin: Stated according to the original source

(i)

Is $A(\alpha )$ an element of $SO(2)$ ?

(ii)

Consider the transformation

\begin{pmatrix} x_{1}' \\ x_{2}' \end{pmatrix} = A(\alpha ) \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix}.

Is $(x_{1}')^{2}+(x_{2}')^{2} = x_{1}^{2}+x_{2}^{2}$ ? Prove or disprove.

(iii)

We define the star product

A(\alpha ) \star A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & 0 & 0 & \sin (\alpha ) \\ 0 & \cos (\alpha ) & \sin (\alpha ) & 0 \\ 0 & \sin (\alpha ) & -\cos (\alpha ) & 0 \\ \sin (\alpha ) & 0 & 0 & -\cos (\alpha ) \end{pmatrix}.

Is $A(\alpha ) \star A(\alpha )$ an element of $SO(4)$ ?

(iv)

Is $A(\alpha ) \otimes A(\alpha )$ an element of $SO(4)$ ?

(v)

Find

X = \left.\frac{dA(\alpha )}{d\alpha }\right|_{\alpha =0}

and then $B(\alpha ) = \mathrm{exp}\left(\alpha X\right)$ . Compare $B(\alpha )$ and $A(\alpha )$ and discuss.

Problem 18.1.38

Given two $2 \times 2$ matrices $A$ and $B$ in the Lie group $GL(2, \mathbb {C})$ , i.e. $\operatorname {det}\left(A\right) \neq 0$ , $\operatorname {det}\left(B\right) \neq 0$ . The action of an $n \times n$ matrix $M = (m_{jk})$ on a polynomial $p(\overrightarrow {x}) \in \mathbf{R}_{n} = \mathbb {C}^{x_{1}, x_{2}, \ldots , x_{n}}$ may be defined by setting

T_{M}p(\overrightarrow {x}) := p(\overrightarrow {x}M)

where $\overrightarrow {x}M$ is the multiplication of the row $n$ -vector $\overrightarrow {x}$ by the $n \times n$ matrix $M$ . Thus $\overrightarrow {x}M$ is a row vector again. We denote by

\mathbf{R}_{4}^{GL(2,\mathbb {C}) \otimes GL(2,\mathbb {C})}

the ring of polynomials in $\mathbf{R}_{4}$ that are invariant under the action of $A \otimes B$ for all pairs $A, B \in GL(2, \mathbb {C})$ , i.e.

\mathbf{R}_{4}^{GL(2,\mathbb {C}) \otimes GL(2,\mathbb {C})} := \left\{ p \in \mathbf{R}_{4} : T_{A \otimes B}p(\overrightarrow {x}) = p(\overrightarrow {x})\right\} .

The action preserves degree and homogeneity. Consider the polynomial

p(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{4}+x_{4}x_{1}

and

A = B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Show that $T_{A \otimes B}p(\overrightarrow {x}) = p(\overrightarrow {x})$ .

Origin: Stated according to the original source

Problem 18.1.39

Let $\alpha , \beta \in \mathbb {R}$ . Then the matrices

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

are elements of the Lie group $SO(2)$ and $A(\alpha ) \otimes B(\beta )$ is an element of the Lie group $SO(4)$ . Find the ``infinitesimal generators'' $X$ and $Y$ such that

\mathrm{exp}\left(\alpha X+\beta Y\right) = A(\alpha ) \otimes B(\beta ).

Origin: Stated according to the original source

Problem 18.1.40

(i)(ii)

A topological group $G$ is both a group and a topological space, the two structures are related by the requirement that the maps $x \mapsto x^{-1}$ (of $G$ onto $G$ ) and $(x, y) \mapsto xy$ (of $G \times G$ onto $G$ ) are continuous. $G \times G$ is given by the product topology.

Origin: Stated according to the original source

(i)

Given a topological group $G$ , define the maps

\phi (x) := xax^{-1}

and

\psi (x) := xax^{-1}a^{-1} \equiv [x, a].

How are the iterates of the maps $\phi$ and $\psi$ related?

(ii)

Consider $G = SO(2)$ and

x = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}, \qquad a = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

with $x, a \in SO(2)$ . Calculate $\phi$ and $\psi$ . Discuss.

Problem 18.1.41

The unitary matrices are elements of the Lie group $U(n)$ . The corresponding Lie algebra $u(n)$ is the set of matrices with the condition $X^{*} = -X$ . An important subgroup of $U(n)$ is the Lie group $SU(n)$ with the condition that $\operatorname {det}\left(U\right) = 1$ . The unitary matrices

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

are not elements of the Lie group $SU(2)$ since the determinants of these unitary matrices are $-1$ . The corresponding Lie algebra $su(n)$ of the Lie group $SU(n)$ are the $n \times n$ matrices given by

X^{*} = -X, \qquad \operatorname {tr}(X) = 0.

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Then any unitary matrix in $U(2)$ can be represented by

U(\alpha , \beta , \gamma , \delta ) = e^{i\alpha I_{2}}e^{-i\beta \sigma _{3}/2}e^{-i\gamma \sigma _{2}/2}e^{-i\delta \sigma _{3}/2}

where $0 \leq \alpha < 2\pi$ , $0 \leq \beta < 2\pi$ , $0 \leq \gamma \leq \pi$ and $0 \leq \delta < 2\pi$ . Calculate the right-hand side.

Origin: Stated according to the original source

Problem 18.1.42

We consider the following subgroups of the Lie group $SL(2, \mathbb {R})$ . Let

K := \left\{ \begin{pmatrix} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \end{pmatrix} : \theta \in [0, 2\pi )\right\}

A := \left\{ \begin{pmatrix} r^{1/2} & 0 \\ 0 & r^{-1/2} \end{pmatrix} : r > 0\right\}

N := \left\{ \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} : t \in \mathbb {R}\right\} .

Any matrix $m \in SL(2, \mathbb {R})$ can be written in a unique way as the product $m = kan$ with $k \in K$ , $a \in A$ and $n \in N$ . This decomposition is called Iwasawa decomposition and has a natural generalization to $SL(n, \mathbb {R})$ , $n \geq 3$ . The notation of the subgroups comes from the fact that $K$ is a compact subgroup, $A$ is an abelian subgroup and $N$ is a nilpotent subgroup of $SL(2, \mathbb {R})$ . Find the Iwasawa decomposition of the matrix

\begin{pmatrix} \sqrt{2} & 1 \\ 1 & \sqrt{2} \end{pmatrix}.

Origin: Stated according to the original source

Problem 18.1.43

(i)(ii)

Let $0 \leq \alpha < \pi /4$ . Consider the transformation

X(x, y, \alpha ) = \frac{1}{\sqrt{\cos (2\alpha )}}(x\cos (\alpha )+iy\sin (\alpha )),

Y(x, y, \alpha ) = \frac{1}{\sqrt{\cos (2\alpha )}}(-ix\sin (\alpha )+y\cos (\alpha )).

Origin: Stated according to the original source

(i)

Show that $X^{2}+Y^{2} = x^{2}+y^{2}$ .

(ii)

Do the matrices

\frac{1}{\sqrt{\cos (2\alpha )}} \begin{pmatrix} \cos (\alpha ) & i\sin (\alpha ) \\ -i\sin (\alpha ) & \cos (\alpha ) \end{pmatrix}

form a group under matrix multiplication?

Problem 18.1.44

(i)(ii)

The Schur-Weyl duality theorem states that if the natural representation of the unitary group $U(N)$ (or of the general linear group $GL(n, \mathbb {C})$ ) is tensored (Kronecker product) $n$ times, then under the joint action of the symmetry group $S_{n}$ (permutation matrices) and $GL(n, \mathbb {C})$ this tensor product is decomposed into a direct sum of multiplicity-free irreducible subrepresentations uniquelly labelled by Young diagrams. Let $n = 4$ . Consider the permutation matrix (flip matrix)

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

and the Hadamard matrix

U_{H} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

which is an element of the Lie group $U(2)$ . Then

U_{H} \otimes U_{H} = \frac{1}{2} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{pmatrix}

which is an element of the Lie group $U(4)$ . Let

\overrightarrow {v}_{1} = \begin{pmatrix} v_{1,1} \\ v_{1,2} \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \begin{pmatrix} v_{2,1} \\ v_{2,2} \end{pmatrix}

be elements of $\mathbb {C}^{2}$ .

Origin: Stated according to the original source

(i)

Find the commutator $[P, U_{H} \otimes U_{H}]$ .

(ii)

Find $P(U_{H} \otimes U_{H})(\overrightarrow {v}_{1} \otimes \overrightarrow {v}_{2})$ . Discuss.

Problem 18.1.45

(i)(ii)

The octonion algebra $\mathcal{O}$ is an 8-dimensional non-associative algebra. It is defined in terms of the basis elements $e_{\mu }$ ( $\mu = 0, 1, \ldots , 7$ ) and their multiplication table. $e_{0}$ is the unit element. We use Greek indices $(\mu , \nu , \ldots )$ to include the 0 and latin indices $(i, j, k, \ldots )$ when we exclude the 0. We define

\hat{e}_{k} := e_{4+k} \quad \text{for} \quad k = 1, 2, 3.

The multiplication rules among the basis elements of octonions $e_{\mu }$ are given by

e_{i}e_{j} = -\delta _{ij}e_{0} + \sum _{k=1}^{3} \epsilon _{ijk}e_{k}, \qquad i, j, k = 1, 2, 3 \tag {1}

and

-e_{4}e_{i} = e_{i}e_{4} = \hat{e}_{i}, \qquad e_{4}\hat{e}_{i} = -\hat{e}_{i}e_{4} = e_{i}, \qquad e_{4}e_{4} = -e_{0}

\hat{e}_{i}\hat{e}_{j} = -\delta _{ij}e_{0} - \sum _{k=1}^{3} \epsilon _{ijk}e_{k}, \qquad i, j, k = 1, 2, 3

-\hat{e}_{j}e_{i} = e_{i}\hat{e}_{j} = -\delta _{ij}e_{4} - \sum _{k=1}^{3} \epsilon _{ijk}\hat{e}_{k}, \qquad i, j, k = 1, 2, 3

where $\delta_{ij}$ is the Kronecker delta and $\epsilon_{ijk}$ is $+1$ if $(ijk)$ is an even permutation of $(123)$ , $-1$ if $(ijk)$ is an odd permutation of $(123)$ and 0 otherwise. We can formally summarize the multiplications as

e_{\mu }e_{\nu } = g_{\mu \nu }e_{0} + \sum _{k=1}^{7} \gamma _{\mu \nu }^{k}e_{k}

where

g_{\mu \nu } = \operatorname {diag}(1, -1, -1, -1, -1, -1, -1, -1), \qquad \gamma _{ij}^{k} = -\gamma _{ji}^{k}

with $\mu , \nu = 0, 1, \ldots , 7$ , and $i, j, k = 1, 2, \ldots , 7$ .

Origin: Stated according to the original source

(i)

Show that the set $\left\{ e_{0}, e_{1}, e_{2}, e_{3}\right\}$ is a closed associative subalgebra.

(ii)

Show that the octonian algebra $\mathcal{O}$ is non-associative.

Section 18.2Supplementary Problems[44/0%]

Problem 18.2.1

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

Origin: Stated according to the original source

(i)

Let $I_{2}, 0_{2}$ be the $2 \times 2$ identity and zero matrix, respectively. Find the group generated by the $4 \times 4$ matrix

M = \begin{pmatrix} 0_{2} & -I_{2} \\ I_{2} & 0_{2} \end{pmatrix}.

(ii)

Find the group generated by the $4 \times 4$ matrix

X = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \otimes \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

Find the eigenvalues of the matrix.

(iii)

Find the group generated by the $6 \times 6$ matrix

B = \begin{pmatrix} 0_{2} & 0_{2} & I_{2} \\ I_{2} & 0_{2} & 0_{2} \\ 0_{2} & I_{2} & 0_{2} \end{pmatrix}

under matrix multiplication.

(iv)

Find the group (matrix multiplication) generated by the $4 \times 4$ matrix

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

Is the group commutative? Find the determinant of all these matrices from (i). Do these numbers form a group under multiplication? Find all the eigenvalues of these matrices. Do these numbers form a group under multiplication? Let $\alpha \in \mathbb {R}$ . Find $\mathrm{exp}\left(\alpha A\right)$ . Is $\mathrm{exp}\left(\alpha A\right)$ an element of $SL(4, \mathbb {R})$ ?

Problem 18.2.2

(i)(ii)

Origin: Stated according to the original source

(i)

What group is generated by the two $3 \times 3$ permutation matrices

A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}

under matrix multiplication? Note that $A^{2} = B^{2} = I_{3}$ .

(ii)

Find the group generated by the $4 \times 4$ permutation matrices

P_{1} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}, \qquad P_{2} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}

under matrix multiplication. There are twenty-four $4 \times 4$ permutation matrices which form a group under matrix multiplication.

Problem 18.2.3

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

Origin: Stated according to the original source

(i)

Let $\alpha , \beta \in \mathbb {R}$ and $\beta \neq 0$ . Do the $2 \times 2$ matrices

M(\alpha , \beta ) = \begin{pmatrix} \cos (\alpha ) & \beta ^{-1}\sin (\alpha ) \\ -\beta \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}

form a group under matrix multiplication? Note that $\operatorname {det}\left(M(\alpha , \beta )\right) = 1$ .

(ii)

Let $\alpha , \beta , \phi \in \mathbb {R}$ and $\alpha , \beta \neq 0$ . Consider the $2 \times 2$ matrices

A(\alpha , \beta , \phi ) = \begin{pmatrix} \alpha \cos (\phi ) & -\beta \sin (\phi ) \\ \beta ^{-1}\sin (\phi ) & \alpha ^{-1}\cos (\phi ) \end{pmatrix}.

Do the matrices form a group under matrix multiplication?

(iii)

Let $\alpha , \theta \in \mathbb {R}$ . Do the matrices

\begin{pmatrix} \cosh (\alpha ) & e^{i\theta }\sinh (\alpha ) \\ e^{-i\theta }\sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix}

form a group under matrix multiplication? Are the matrices unitary?

(iv)

Let $\alpha , \beta , \tau \in \mathbb {R}$ . Do the $2 \times 2$ matrices

\begin{pmatrix} e^{i(\alpha +\beta )}\cosh (\tau ) & e^{i(\alpha -\beta )}\sinh (\tau ) \\ e^{-i(\alpha -\beta )}\sinh (\tau ) & e^{-i(\alpha +\beta )}\cosh (\tau ) \end{pmatrix}

form a group under matrix multiplication?

Problem 18.2.4

Do the eight $2 \times 2$ matrices

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

\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}

form a group under matrix multiplication? If not add the matrices so that one has a group.

Origin: Stated according to the original source

Problem 18.2.5

(i)(ii)

Origin: Stated according to the original source

(i)

Do the $4 \times 4$ matrices

g(\phi , \theta ) = \begin{pmatrix} e^{i\phi }\cosh (\theta ) & 0 & 0 & \sinh (\theta ) \\ 0 & e^{-i\phi }\cosh (\theta ) & \sinh (\theta ) & 0 \\ 0 & \sinh (\theta ) & e^{i\phi }\cosh (\theta ) & 0 \\ \sinh (\theta ) & 0 & 0 & e^{-i\phi }\cosh (\theta ) \end{pmatrix}

form a group under matrix multiplication?

(ii)

Do the $4 \times 4$ matrices

g(\phi , \theta ) = \begin{pmatrix} e^{i\phi }\cos (\theta ) & 0 & 0 & -\sin (\theta ) \\ 0 & e^{-i\phi }\cos (\theta ) & -\sin (\theta ) & 0 \\ 0 & \sin (\theta ) & e^{i\phi }\cos (\theta ) & 0 \\ \sin (\theta ) & 0 & 0 & e^{-i\phi }\cos (\theta ) \end{pmatrix}

form a group under matrix multiplication?

Problem 18.2.6

The Lie group $O(2)$ is generated by a rotation $R_{1}$ and a reflection $R_{2}$

R_{1}(\theta ) = \begin{pmatrix} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \end{pmatrix}, \qquad R_{2}(\theta ) = \begin{pmatrix} \cos (\theta ) & \sin (\theta ) \\ \sin (\theta ) & -\cos (\theta ) \end{pmatrix}.

Show that

\operatorname {tr}(R_{1}(\theta )) = 2\cos (\theta ), \quad \operatorname {tr}(R_{2}(\theta )) = 0, \quad \operatorname {det}\left(R_{1}(\theta )\right) = 1, \quad \operatorname {det}\left(R_{2}(\theta )\right) = -1

and $\operatorname {det}\left(R_{1}(\theta )R_{2}(\theta )\right) = -1$ , $\operatorname {tr}(R_{1}(\theta )R_{2}(\theta )) = 0$ .

Origin: Stated according to the original source

Problem 18.2.7

(i)(ii)

Origin: Stated according to the original source

(i)

Find all $3 \times 3$ permutation matrices $P$ such that

P^{-1} \begin{pmatrix} 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & 1 & 0 \\ 1/\sqrt{2} & 0 & -1/\sqrt{2} \end{pmatrix} P = \begin{pmatrix} 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & 1 & 0 \\ 1/\sqrt{2} & 0 & -1/\sqrt{2} \end{pmatrix}.

Show that these matrices form a group, i.e. a subgroup of the $3 \times 3$ permutation matrices.

(ii)

Find all $2 \times 2$ invertible matrices $B$ with $\operatorname {det}\left(B\right) = 1$ such that

B \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} B^{-1} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

Show that these matrices form a group under matrix multiplication.

Problem 18.2.8

Prove or disprove: For any permutation matrix their eigenvalues form a group under multiplication.

Origin: Stated according to the original source

Problem 18.2.9

Let $\sigma_{j}$ ( $j = 0, 1, 2, 3$ ) be the Pauli spin matrices, where $\sigma_{0} = I_{2}$ . Do the $4 \times 4$ matrices

\begin{pmatrix} \sigma _{j} & 0_{2} \\ 0_{2} & \sigma _{k} \end{pmatrix}, \qquad \begin{pmatrix} 0_{2} & \sigma _{j} \\ \sigma _{k} & 0_{2} \end{pmatrix}, \qquad j, k = 0, 1, 2, 3

form a group under matrix multiplication? If not add the elements to find a group.

Origin: Stated according to the original source

Problem 18.2.10

Let $\omega := \mathrm{exp}\left(2\pi i/4\right)$ . Consider the $3 \times 3$ unitary matrices

\Sigma = \begin{pmatrix} \omega & 0 & 0 \\ 0 & \omega ^{2} & 0 \\ 0 & 0 & \omega ^{3} \end{pmatrix}, \qquad C = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix},

\Omega = \begin{pmatrix} -i/2 & (1+i)/2 & 1/2 \\ (1+i)/2 & 0 & (1-i)/2 \\ 1/2 & (1-i)/2 & i/2 \end{pmatrix}.

Do the matrices of the set

\Lambda := \left\{ \Sigma ^{j}C^{k}\Omega ^{\ell } : 0 \leq j \leq 3, \ 0 \leq k \leq 1, \ 0 \leq \ell \leq 2\right\}

form a group under matrix multiplication?

Origin: Stated according to the original source

Problem 18.2.11

Consider the set of $2 \times 2$ matrices

\left\{ e = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad a = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\right\} .

Then under matrix multiplication we have a group. Consider the set

\left\{ e \otimes e, \quad e \otimes a, \quad a \otimes e, \quad a \otimes a\right\} .

Show that this set forms a group under matrix multiplication. For example we have $(a \otimes a)(a \otimes a) = e \otimes e$ . The neutral element is $e \otimes e$ .

Origin: Stated according to the original source

Problem 18.2.12

Consider the $2n \times 2n$ matrix

J := \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix}.

We define that the $2n \times 2n$ matrix $H$ over $\mathbb {R}$ is Hamiltonian if $(JH)^{T} = JH$ . We define that the $2n \times 2n$ matrix $S$ over $\mathbb {R}$ is symplectic if $S^{T}JS = J$ . Show that if $H$ is Hamiltonian and $S$ is symplectic, then the matrix $S^{-1}HS$ is Hamiltonian. Note that since $S$ invertible $S^{T}$ is invertible. We have $(JS^{-1}HS)^{T} = S^{T}H^{T}(S^{T})^{-1}J^{T}$ .

Origin: Stated according to the original source

Problem 18.2.13

(i)(ii)

Let $\alpha , \beta , \gamma \in \mathbb {R}$ . The Heisenberg group $H_{3}(\mathbb {R})$ consists of all $3 \times 3$ upper triangle matrices of the form

M(\alpha , \beta , \gamma ) = \begin{pmatrix} 1 & \alpha & \gamma \\ 0 & 1 & \beta \\ 0 & 0 & 1 \end{pmatrix}

with matrix multiplication as composition. Let $t \in \mathbb {R}$ . Consider the matrices

A(t) = \begin{pmatrix} 1 & t & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad B(t) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{pmatrix}, \qquad C(t) = \begin{pmatrix} 1 & 0 & t \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Show that $\left\{ A(t) : t \in \mathbb {R}\right\}$ , $\left\{ B(t) : t \in \mathbb {R}\right\}$ , $\left\{ C(t) : t \in \mathbb {R}\right\}$ are one-parameter subgroups in $H_{3}(\mathbb {R})$ .

(ii)

Show that $\left\{ C(t) : t \in \mathbb {R}\right\}$ is the center of $H_{3}(\mathbb {R})$ .

Problem 18.2.14

Let $\omega^{3} = 1$ . What group is generated by the $3 \times 3$ matrices

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

under matrix multiplication? Is the matrix $\Pi = \frac{1}{3}(A+B+C)$ a projection matrix?

Origin: Stated according to the original source

Problem 18.2.15

Consider the six $3 \times 3$ permutation matrices denoted by $P_{0}, P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ , where $P_{0}$ is the $3 \times 3$ identity matrix. Find all triples $(Q_{1}, Q_{2}, Q_{3})$ of these permutation matrices such that $Q_{1}Q_{2}Q_{3} = Q_{3}Q_{2}Q_{1}$ , where $Q_{1} \neq Q_{2}$ , $Q_{2} \neq Q_{3}$ , $Q_{3} \neq Q_{1}$ .

Origin: Stated according to the original source

Problem 18.2.16

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

Let $n \geq 2$ and $\omega = \mathrm{exp}\left(2\pi i/n\right)$ , where $\omega$ is the $n$ -th primitive root of unity with $\omega^{n} = 1$ . Consider the diagonal and permutation matrices, respectively

D = \begin{pmatrix} 1 & 0 & 0 & & 0 \\ 0 & \omega & 0 & & 0 \\ 0 & 0 & \omega ^{2} & & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & \omega ^{n-1} \end{pmatrix}, \qquad P = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ 1 & 0 & 0 & \cdots & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Show that $D^{n} = P^{n} = I_{n}$ .

(ii)

Show that the set of matrices $\left\{ D^{j}P^{k} : j, k = 0, 1, \ldots , n-1\right\}$ form a basis of the vector space of $n \times n$ matrices.

(iii)

Show that $PD = \omega DP$ , $P^{j}D^{k} = \omega^{jk}D^{k}P^{j}$ .

(iv)

Find the matrix $X = \zeta P + \zeta^{-1}P^{-1} + \eta D + \eta^{-1}D^{-1}$ and calculate the eigenvalues.

(v)

Let $U$ be the unitary matrix

U = \frac{1}{\sqrt{n}}(\omega ^{jk})

where $j, k = 0, 1, \ldots , n-1$ . Show that $UPU^{-1} = D$ .

(vi)

Let $R = P \otimes I_{n}$ and $S = D \otimes I_{n}$ . Find $RS$ . Let $X = P \otimes P$ and $Y = D \otimes D$ . Find $XY$ . Find the commutator $[X, Y]$ .

Problem 18.2.17

(i)(ii)

Consider the diagonal matrix $D$ and the permutation matrix $P$

D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \mathrm{exp}\left(2i\pi /3\right) & 0 \\ 0 & 0 & \mathrm{exp}\left(-2i\pi /3\right) \end{pmatrix}, \qquad P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

What group is generated by $D$ and $P$ ?

(ii)

Calculate the commutator $[D, P]$ . Discuss.

Problem 18.2.18

(i)(ii)

In the Lie group $U(N)$ of the $N \times N$ unitary matrices one can find two $N \times N$ matrices $U$ and $V$ such that

UV = e^{2\pi i/N}VU.

Any $N \times N$ hermitian matrix $H$ can be written in the form

H = \sum _{j=0}^{N-1} \sum _{k=0}^{N-1} h_{jk}U^{j}V^{k}.

Using the expansion coefficients $h_{jk}$ one can associate to the hermitian matrix $H$ the function

h(q,q) = \sum _{j=0}^{N-1} \sum _{k=1}^{N-1} h_{jk}e^{2\pi i(jp+kq)}

where $p = 0, 1, \ldots , N-1$ and $q = 0, 1, \ldots , N-1$ . Consider the case $N = 2$ and

U = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi } \end{pmatrix}, \qquad V = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Origin: Stated according to the original source

(i)

Consider the hermitian and unitary matrix

\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

Find $h(p, q)$ .

(ii)

Consider the hermitian and projection matrix

\frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.

Find $h(p, q)$ .

Problem 18.2.19

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the $2 \times 2$ matrix

J := -i\sigma _{2} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.

Show that any $2 \times 2$ matrix $A \in SL(2, \mathbb {C})$ satisfies $A^{T}JA = J$ .

(ii)

Let $A$ satisfying $A^{T}JA = J$ . Is $(A \otimes A)^{T}(J \otimes J)(A \otimes A) = J \otimes J$ ? Is $(A \oplus A)^{T}(J \oplus J)(A \oplus A) = J \oplus J$ ? Is $(A \star A)^{T}(J \star J)(A \star A) = J \star J$ ?

Problem 18.2.20

The group $SL(2, \mathbb {F}_{3})$ consists of unimodular $2 \times 2$ matrices with integer entries taken modulo three. Let

A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}

be an element of $SL(2, \mathbb {F}_{3})$ . Find the inverse of $A$ .

Origin: Stated according to the original source

Problem 18.2.21

Consider the semi-simple Lie group $SL(2, \mathbb {R})$ . Let $A \in SL(2, \mathbb {R})$ . Then $A^{-1} \in SL(2, \mathbb {R})$ . Explain why. Show that $A$ and $A^{-1}$ have the same eigenvalues. Is this still true for $A \in SL(3, \mathbb {R})$ ?

Origin: Stated according to the original source

Problem 18.2.22

(i)(ii)

Let $A, B \in SL(2, \mathbb {R})$ .

Origin: Stated according to the original source

(i)

Is $\operatorname {tr}(A) = \operatorname {tr}(A^{-1})$ ? Prove or disprove.

(ii)

Is $\operatorname {tr}(AB) = \operatorname {tr}(A)\operatorname {tr}(B) - \operatorname {tr}(AB^{-1})$ ? Prove or disprove.

Problem 18.2.23

(i)(ii)

Let $\alpha \in \mathbb {R}$ . Consider the $2 \times 2$ matrix

A(\alpha ) = \frac{1}{\sqrt{2}} \begin{pmatrix} e^{\alpha } & -e^{\alpha } \\ e^{-\alpha } & e^{-\alpha } \end{pmatrix}.

Origin: Stated according to the original source

(i)

Find the trace, determinant, eigenvalues and normalized eigenvectors of the matrix $A(\alpha )$ .

(ii)

Calculate

X := \left.\frac{dA(\alpha )}{d\alpha }\right|_{\alpha =0}.

Find $\mathrm{exp}\left(\alpha X\right)$ and compare with $A(\alpha )$ , i.e. is $\mathrm{exp}\left(\alpha X\right) = A(\alpha )$ ? Discuss.

Problem 18.2.24

(i)(ii)(iii)

Origin: Stated according to the original source

(i)

Let $\alpha \in \mathbb {R}$ . The $2 \times 2$ rotation matrix

R(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}

is an element of the Lie group $SO(2, \mathbb {R})$ . Find the spectral decomposition of $R(\alpha )$ .

(ii)

Let $\beta \in \mathbb {R}$ . The matrix

B(\beta ) = \begin{pmatrix} \cosh (\beta ) & \sinh (\beta ) \\ \sinh (\beta ) & \cosh (\beta ) \end{pmatrix}

is an element of the Lie group $SO(1, 1, \mathbb {R})$ . Find the spectral decomposition of $B(\beta )$ .

(iii)

Find the spectral decomposition of $R(\alpha ) \otimes B(\beta )$ .

Problem 18.2.25

The Lie group $SU(1,1)$ consists of all $2 \times 2$ matrices $T$ over the complex numbers with

TMT^{*} = M, \qquad M = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad \operatorname {det}\left(T\right) = 1.

Show that the conditions on $\xi_{0}, \xi_{1}, \xi_{2}, \xi_{3} \in \mathbb {R}$ such that

T = \begin{pmatrix} \xi _{0}+i\xi _{3} & \xi _{1}+i\xi _{2} \\ \xi _{1}-i\xi _{2} & \xi _{0}-i\xi _{3} \end{pmatrix}

is an element of $SU(1,1)$ is given by $\operatorname {det}\left(T\right) = \xi_{0}^{2}+\xi_{3}^{2}-\xi_{1}^{2}-\xi_{2}^{2} = 1$ .

Origin: Stated according to the original source

Problem 18.2.26

(i)(ii)

The unit sphere

S^{3} := \left\{ (x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb {R}^{4} : \sum _{j=1}^{4} x_{j}^{2} = 1\right\}

we identify with the Lie group $SU(2)$ via the map

(x_{1}, x_{2}, x_{3}, x_{4}) \mapsto \begin{pmatrix} x_{1}+ix_{2} & -x_{3}+ix_{4} \\ x_{3}+ix_{4} & x_{1}-ix_{2} \end{pmatrix}.

Origin: Stated according to the original source

(i)

Map the standard basis of $\mathbb {R}^{4}$ into $SU(2)$ and express these matrices using the Pauli spin matrices and the $2 \times 2$ identity matrix.

(ii)

Map the Bell basis

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

into $SU(2)$ and express these matrices using the Pauli spin matrices and the $2 \times 2$ identity matrix.

Problem 18.2.27

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Show that if $A^{T} = -A$ , then $e^{A} \in O(n, \mathbb {C})$ .

Origin: Stated according to the original source

Problem 18.2.28

(i)(ii)

Origin: Stated according to the original source

(i)

Can any element of the Lie group $SU(1,1)$ be written as

\begin{pmatrix} e^{i\alpha /2} & 0 \\ 0 & e^{-i\alpha /2} \end{pmatrix} \begin{pmatrix} \cosh (\zeta /2) & \sinh (\zeta /2) \\ \sinh (\zeta /2) & \cosh (\zeta /2) \end{pmatrix} \begin{pmatrix} e^{i\beta /2} & 0 \\ 0 & e^{-i\beta /2} \end{pmatrix}?

Each element in the this product is an element of the Lie group $SU(1,1)$ .

(ii)

Can any element of the compact Lie group $SU(2)$ be written as

\begin{pmatrix} \cos (\alpha /2) & -\sin (\alpha /2) \\ \sin (\alpha /2) & \cos (\alpha /2) \end{pmatrix} \begin{pmatrix} e^{i\gamma /2} & 0 \\ 0 & e^{-i\gamma /2} \end{pmatrix} \begin{pmatrix} \cos (\beta /2) & \sin (\beta /2) \\ -\sin (\beta /2) & \cos (\beta /2) \end{pmatrix}?

Problem 18.2.29

Consider the Lie group $SL(2, \mathbb {R})$ , i.e. the set of all real $2 \times 2$ matrices with determinant equal to 1. A dynamical system in $SL(2, \mathbb {R})$ can be defined by

M_{k+2} = M_{k}M_{k+1}, \qquad k = 0, 1, 2, \ldots

with the initial matrices $M_{0}, M_{1} \in SL(2, \mathbb {R})$ . Let $F_{k} := \operatorname {tr}(M_{k})$ . Is

F_{k+3} = F_{k+2}F_{k+1}-F_{k} \qquad k = 0, 1, 2, \ldots ?

Prove or disprove. Use that property that for any $2 \times 2$ matrix $A$ we have (Cayley-Hamilton theorem)

A^{2}-A\operatorname {tr}(A)+I_{2}\operatorname {det}\left(A\right) = 0_{2}.

Origin: Stated according to the original source

Problem 18.2.30

(i)(ii)

Origin: Stated according to the original source

(i)

Let $\alpha \in \mathbb {R}$ . Do the matrices

A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & i\sin (\alpha ) \\ i\sin (\alpha ) & \cos (\alpha ) \end{pmatrix}

form a group under matrix multiplication? Note that $A(0) = I_{2}$ . So we have the neutral element. Furthermore $\operatorname {det}\left(A(\alpha )\right) = 1$ .

(ii)

Let $\alpha \in \mathbb {R}$ . Do the matrices

B(\alpha ) = \begin{pmatrix} \cosh (\alpha ) & i\sinh (\alpha ) \\ -i\sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix}

form a group under matrix multiplication? Note $B(0) = I_{2}$ . So we have a neutral element. Furthermore $\operatorname {det}\left(B(\alpha )\right) = 1$ .

Problem 18.2.31

(i)(ii)(iii)

Consider the $3 \times 3$ matrix

A(\theta ) = \begin{pmatrix} \cos (\theta ) & 0 & \sin (\theta ) \\ 0 & 1 & 0 \\ -\sin (\theta ) & 0 & \cos (\theta ) \end{pmatrix}.

Origin: Stated according to the original source

(i)

Show that $A(\theta )$ is an element of the compact Lie group $SO(3, \mathbb {R})$ .

(ii)

Show that the eigenvalues of $A(\theta )$ are $\lambda_{1} = 1$ , $\lambda_{2} = e^{i\theta }$ , $\lambda_{3} = e^{-i\theta }$ .

(iii)

Show that the $3 \times 3$ matrix

O(\theta , \phi ) = \begin{pmatrix} \sin (\theta )\cos (\phi ) & -\sin (\phi ) & -\cos (\theta )\cos (\phi ) \\ \sin (\theta )\sin (\phi ) & \cos (\phi ) & \cos (\theta )\sin (\phi ) \\ \cos (\theta ) & 0 & \sin (\theta ) \end{pmatrix}

is an element of the compact Lie group $SO(3)$ .

Problem 18.2.32

Let $n$ be an integer with $n \geq 2$ . Let $p, q$ be integers with $p, q \geq 1$ and $n = p+q$ . Let $I_{p}$ be the $p \times p$ identity matrix and $I_{q}$ be the $q \times q$ identity matrix. Let

I_{p,q} = \begin{pmatrix} I_{p} & 0 \\ 0 & -I_{q} \end{pmatrix} \equiv I_{p} \oplus (-I_{q}).

The Lie group $O(p,q)$ is the set of all $n \times n$ matrices defined by

O(p,q) := \left\{ A \in GL(n, \mathbb {R}) : A^{T}I_{p,q}A = I_{p,q}\right\} .

Show that this is the group of all invertible linear maps of $\mathbb {R}^{n}$ that preserves the quadratic form

\sum _{j=1}^{p} x_{j}y_{j} - \sum _{j=p+1}^{n} x_{j}y_{j}.

Origin: Stated according to the original source

Problem 18.2.33

The Lie group $SU(2,2)$ is defined as the group of transformation on the four dimensional complex space $\mathbb {C}^{4}$ leaving invariant the indefinite quadratic form

\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}-\left|z_{3}\right|^{2}-\left|z_{4}\right|^{2}.

The Lie algebra $su(2,2)$ is defined as the $4 \times 4$ matrices $X$ with trace 0 and $X^{*}L+LX = 0_{4}$ , where $L$ is the $4 \times 4$ matrix

L = \begin{pmatrix} -I_{2} & 0_{2} \\ 0_{2} & I_{2} \end{pmatrix}.

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

an element of the Lie algebra $su(2,2)$ ? Find $\mathrm{exp}\left(zX\right)$ . Discuss.

Origin: Stated according to the original source

Problem 18.2.34

(i)(ii)(iii)

Let $\theta \in \mathbb {R}$ .

Origin: Stated according to the original source

(i)

Is the $2 \times 2$ matrix

M_{1}(\theta ) = \begin{pmatrix} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \end{pmatrix} \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}

unitary? Prove or disprove. If so is the matrix an element of the Lie group $SU(2)$ ?

(ii)

Is the $8 \times 8$ matrix

M_{2}(\theta ) = \begin{pmatrix} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \end{pmatrix} \otimes \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}

unitary? Prove or disprove. If so is the matrix an element of the Lie group $SU(4)$ ?

(iii)

Let $\oplus$ be the direct sum. Is the $6 \times 6$ matrix

M_{3}(\theta ) = \begin{pmatrix} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \end{pmatrix} \oplus \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \oplus \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}

unitary? Prove or disprove. If so is the matrix an element of the Lie group $SU(4)$ ?

Problem 18.2.35

(i)(ii)(iii)

Write down two $2 \times 2$ matrices $A$ and $B$ which are elements of the Lie group $O(2, \mathbb {R})$ but not elements of $SO(2, \mathbb {R})$ .

Origin: Stated according to the original source

(i)

Is $AB$ an element of the Lie group $SO(2, \mathbb {R})$ ?

(ii)

Is $A \otimes B$ an element of the Lie group $SO(4, \mathbb {R})$ ?

(iii)

Is $A \oplus B$ an element of $SO(4, \mathbb {R})$ ?

Problem 18.2.36

(i)(ii)

Let $z = x+iy$ with $x, y \in \mathbb {R}$ . Consider the $2 \times 2$ matrix

\begin{pmatrix} \cos (z) & -\sin (z) \\ \sin (z) & \cos (z) \end{pmatrix}.

One has the identities

\cos (x+iy) \equiv \cos (x)\cos (iy)-\sin (x)\sin (iy) \equiv \cos (x)\cosh (y)-i\sin (x)\sinh (y)

\sin (x+iy) \equiv \sin (x)\cos (iy)+\cos (x)\sin (iy) \equiv \sin (x)\cosh (y)+i\cos (x)\sinh (y).

Let $x = 0$ . Then we arrive at the matrix

M(y) = \begin{pmatrix} \cosh (y) & -i\sinh (y) \\ i\sinh (y) & \cosh (y) \end{pmatrix}.

Origin: Stated according to the original source

(i)

Do these matrices ( $y \in \mathbb {R}$ ) form a group under matrix multiplication?

(ii)

Calculate

X = \left.\frac{d}{dy}M(y)\right|_{y=0}

and then $\mathrm{exp}\left(yX\right)$ with $y \in \mathbb {R}$ . Discuss.

Problem 18.2.37

The maximal compact Lie subgroup of $SL(2, \mathbb {C})$ is $SU(2)$ . Give a $2 \times 2$ matrix $A$ which is an element of $SL(2, \mathbb {C})$ , but not an element of $SU(2)$ .

Origin: Stated according to the original source

Problem 18.2.38

Let $A_{1}, A_{2}, \ldots , A_{n}$ be $m \times m$ matrices. We define

\sigma (A_{1}, \ldots , A_{n}) := \frac{1}{n!} \sum _{s \in S_{n}} A_{s(1)} \otimes \cdots \otimes A_{s(n)}

where $S_{n}$ is the permutation group. Let $n = 3$ and consider the matrices

A_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad A_{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad A_{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Find

\sigma (A_{1}, A_{2}, A_{3}) = \frac{1}{3!} \sum _{s \in S_{3}} A_{s(1)} \otimes A_{s(2)} \otimes A_{s(3)}.

Origin: Stated according to the original source

Problem 18.2.39

Let $\alpha , \beta , \gamma \in \mathbb {R}$ . Do the $3 \times 3$ matrices

M(\alpha , \beta , \gamma ) = \begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) & \beta \\ -\sin (\alpha ) & \cos (\alpha ) & \gamma \\ 0 & 0 & 1 \end{pmatrix}

form a group under matrix multiplication?

Origin: Stated according to the original source

Problem 18.2.40

(i)(ii)

Origin: Stated according to the original source

(i)

Consider the non-compact Lie group $SO(1, 1, \mathbb {R})$ with the element

A(\alpha ) = \begin{pmatrix} \cosh (\alpha ) & \sinh (\alpha ) \\ \sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix}.

Show that the inverse of $A(\alpha )$ is given by

A^{-1}(\alpha ) = A(-\alpha ) = \begin{pmatrix} \cosh (\alpha ) & -\sinh (\alpha ) \\ -\sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix}.

Show that the eigenvalues of $A(\alpha )$ are given by $e^{\alpha }$ and $e^{-\alpha }$ .

(ii)

Let $\oplus$ be the direct sum. Find the determinant, eigenvalues and normalized eigenvectors of the $3 \times 3$ matrix $(1) \oplus A(\alpha )$ .

Problem 18.2.41

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

Origin: Stated according to the original source

(i)

Let $A, B$ be elements of $SL(n, \mathbb {R})$ . Show that $A \otimes B$ is an element of $SL(n^{2}, \mathbb {R})$ .

(ii)

Let $A, B$ be elements of $SL(n, \mathbb {R})$ . Show that $A \oplus B$ is an element of $SL(2n, \mathbb {R})$ .

(iii)

Let $A, B$ be elements of $SL(2, \mathbb {R})$ . Let $\star$ be the star product. Show that $A \star B$ an element of $SL(4, \mathbb {R})$ .

(iv)

Are the matrices

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

elements of $SL(3, \mathbb {R})$ and $SL(5, \mathbb {R})$ , respectively?

Problem 18.2.42

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $\left\| A\right\| < 1$ . Then $I_{n}+A$ is invertible ( $I_{n}+A \in GL(n, \mathbb {C})$ ). Can we conclude that $I_{n} \otimes I_{n}+A \otimes A$ is invertible?

Origin: Stated according to the original source

Problem 18.2.43

(i)(ii)

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. We define

\tau _{1} = i\sigma _{1}, \qquad \tau _{2} = i\sigma _{2}, \qquad \tau _{3} = i\sigma _{3}

and $\tau_{0} = I_{2}$ . Thus $\tau_{j}$ ( $j = 0, 1, 2, 3$ ) are elements of the Lie group $SU(2)$ .

Origin: Stated according to the original source

(i)

We define the operator $R$ as ( $k \geq 2$ )

R(\tau _{j_{1}} \otimes \tau _{j_{2}} \otimes \cdots \otimes \tau _{j_{k}}) := \frac{1}{2}(\tau _{j_{1}}\tau _{j_{2}}^{*} \otimes \tau _{j_{2}}\tau _{j_{3}}^{*} \otimes \cdots \otimes \tau _{j_{k}}\tau _{j_{1}}^{*}).

Find $R(\tau_{1} \otimes \tau_{2} \otimes \tau_{3})$ .

(ii)

Consider the unitary matrix (constructed from the four Bell states)

U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & -i \\ 0 & -i & -1 & 0 \\ 0 & -i & 1 & 0 \\ 1 & 0 & 0 & i \end{pmatrix} \Rightarrow U^{*} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & i & i & 0 \\ 0 & -1 & 1 & 0 \\ i & 0 & 0 & -i \end{pmatrix}.

Show that $U(i\sigma_{1} \otimes i\sigma_{1})U$ , $U(i\sigma_{2} \otimes i\sigma_{2})U$ , $U(i\sigma_{3} \otimes i\sigma_{3})U$ are elements of the Lie group $SO(4)$ .

Problem 18.2.44

(i)(ii)

Consider the $8 \times 8$ permutation matrix

P = (1) \oplus \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix} \oplus (1)

and

G = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix}, \qquad \overrightarrow {v}_{1} = \begin{pmatrix} v_{1,1} \\ v_{1,2} \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \begin{pmatrix} v_{2,1} \\ v_{2,2} \end{pmatrix}, \qquad \overrightarrow {v}_{3} = \begin{pmatrix} v_{3,1} \\ v_{3,2} \end{pmatrix}.

Origin: Stated according to the original source

(i)

Show that $[P, G \otimes G \otimes G] = 0_{8}$ , where

G = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

(ii)

Assume that $G$ is the Hadamard matrix. Find the vector

(P(G \otimes G \otimes G))(\overrightarrow {v}_{1} \otimes \overrightarrow {v}_{2} \otimes \overrightarrow {v}_{3}).

Find the eigenvalues and normalized eigenvectors of $P$ . Discuss.