Math Problems Vault

Problem 18.2.1

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

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(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)

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(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)

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(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.

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Problem 18.2.5

(i)(ii)

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

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Problem 18.2.7

(i)(ii)

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(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.

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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.

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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?

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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 $e \otimes e$ .

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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 $(JS^{-1}HS)^{T} = S^{T}H^{T}(S^{T})^{-1}J^{T}$ .

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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}.

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(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?

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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 $Q_{1} \neq Q_{2}$ , $Q_{2} \neq Q_{3}$ , $Q_{3} \neq Q_{1}$ .

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Problem 18.2.16

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}.

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(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}.

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(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}.

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(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)

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(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 $A$ .

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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 $A \in SL(3, \mathbb {R})$ ?

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Problem 18.2.22

(i)(ii)

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

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(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}.

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(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)

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(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 $\operatorname {det}\left(T\right) = \xi_{0}^{2}+\xi_{3}^{2}-\xi_{1}^{2}-\xi_{2}^{2} = 1$ .

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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}.

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(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 $e^{A} \in O(n, \mathbb {C})$ .

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Problem 18.2.28

(i)(ii)

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(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}.

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Problem 18.2.30

(i)(ii)

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(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}.

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(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}.

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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}.

Is

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.

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Problem 18.2.34

(i)(ii)(iii)

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

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(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 $SO(2, \mathbb {R})$ .

?

(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}.

?

(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 $SU(2)$ .

?

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)}.

?

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?

?

Problem 18.2.40

(i)(ii)

?

(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)

?

(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?

?

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 $SU(2)$ .

?

(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}.

?

(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.