Math Problems Vault

Problem 1629 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Subsection 14.2.2 "Operators Preserving the Scalar Product (Isometric Operators)"

Prove that any pseudo-orthogonal matrix of type $(1,1)$ (see problem 1625) has the form

\pm \left(\begin{smallmatrix} \frac{1}{\sqrt{1-\beta ^{2}}} & \pm \frac{\beta }{\sqrt{1-\beta ^{2}}} \\ \frac{\beta }{\sqrt{1-\beta ^{2}}} & \pm \frac{1}{\sqrt{1-\beta ^{2}}} \end{smallmatrix}\right)

(the Lorentz parametrization of the group $O(1,1)$ ).

?

Problem 1630 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Subsection 14.2.2 "Operators Preserving the Scalar Product (Isometric Operators)"

Prove that any pseudo-orthogonal matrix of type $(1,1)$ (see problem 1625) has the form

\pm \left(\begin{smallmatrix} \operatorname {ch} \varphi & \pm \operatorname {sh} \varphi \\ \operatorname {sh} \varphi & \pm \operatorname {ch} \varphi \end{smallmatrix}\right)

(hyperbolic rotations).

?

Problem 4.1.37 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 4.1 "Chapter Problems"

Find all $2 \times 2$ matrices $g$ over $\mathbb {C}$ such that $\operatorname {det}\left(g\right) = 1$ , $\eta g^{*} \eta = g^{-1}$ , where $\eta$ is the diagonal matrix $\eta = \operatorname {diag}(1,-1)$ .

?

Problem 9.1.23 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 9.1 "Chapter Problems"

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

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

S(\zeta ) = \begin{pmatrix} \cosh (\zeta ) & 0 & 0 & \sinh (\zeta ) \\ 0 & \cosh (\zeta ) & \sinh (\zeta ) & 0 \\ 0 & \sinh (\zeta ) & \cosh (\zeta ) & 0 \\ \sinh (\zeta ) & 0 & 0 & \cosh (\zeta ) \end{pmatrix}.

?

(i)

Show that the matrix is invertible, i.e. find the determinant.

(ii)

Calculate the inverse of $S(\zeta )$ .

(iii)

Calculate

A := \frac{d}{d\zeta }S(\zeta )\bigg|_{\zeta =0}

and then calculate $\exp (\zeta A)$ .

(iv)

Do the matrices $S(\zeta )$ form a group under matrix multiplication?

Problem 18.1.17 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.1 "Chapter Problems"

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 $Sp(2n)$ .

?

Problem 18.2.25 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.2 "Supplementary Problems"

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

?

Problem 18.2.28 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.2 "Supplementary Problems"

(i)(ii)

?

(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 18.2.32 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.2 "Supplementary Problems"

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}.

?

Problem 18.2.33 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.2 "Supplementary Problems"

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.

?

Problem 18.2.40 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.2 "Supplementary Problems"

(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 19.2.10 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 19.2 "Supplementary Problems"

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 $\mathfrak {su}(2,2)$ 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 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}

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

?

Problem 2.2 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

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

The Lie group $SU(1,1)$ consists of the set of all $2 \times 2$ pseudo-unitary matrices (of determinant $1$ ) preserving the quadratic form

\left|z_{1}\right|^{2} - \left|z_{2}\right|^{2}, \quad (z_{1}, z_{2} \in \mathbb {C}) .

?

(i)

Show that the $2 \times 2$ matrix

U(\tau , \alpha , \beta ) = \begin{pmatrix} \cosh (\tau /2) e^{-i\alpha } & \sinh (\tau /2) e^{-i\beta } \\ \sinh (\tau /2) e^{i\beta } & \cosh (\tau /2) e^{i\alpha } \end{pmatrix} \tag {1}

preserves the quadratic form $\left|z_{1}\right|^{2} - \left|z_{2}\right|^{2}$ $(z_{1}, z_{2} \in \mathbb {C})$ , where $\tau , \alpha , \beta \in \mathbb {R}$ .

(ii)

Show that $\operatorname {det}\left(U(\tau , \alpha , \beta )\right) = 1$ .

(iii)

Give the inverse of $U(\tau , \alpha , \beta )$ .

(iv)

Show that $U_{1}(\tau_{1}, \alpha_{1}, \beta_{1}) U_{2}(\tau_{2}, \alpha_{2}, \beta_{2})$ with $\alpha_{1} + \alpha_{2} = \beta_{1} - \beta_{2}$ is again a matrix of the form (1).

Problem 2.12 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

Given the invertible $4 \times 4$ matrix

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

Find two non-diagonal matrices $L$ over $\mathbb {R}$ such that $L^{T}GL = G$ . Such transformations are known as Lorentz transformations.

?

Problem 2.38 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

Consider the Lie group $SO(1,1,\mathbb {R})$ . Let

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

be an element of $SO(1,1,\mathbb {R})$ . Find a $2 \times 2$ matrix $B$ such that $B^{-1}A(\phi )B$ is diagonal. To find $B$ calculate the eigenvalues and normalized eigenvectors of $A(\phi )$ . Is $B$ an element of the Lie group $SO(1,1,\mathbb {R})$ ?

?

Problem 2.77 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

One natural parameterization of the Lie algebra $su(1,1)$ as a sub Lie algebra of $sp(2,2)$ is given by the $4 \times 4$ matrices

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

with $[R_{1}, R_{2}] = -2R_{3}$ , $[R_{3}, R_{1}] = 2R_{2}$ , $[R_{2}, R_{3}] = -2R_{1}$ . Find $g_{2}g_{1}g_{2}$ , where $g_{1} = \mathrm{exp}\left(\alpha R_{1}\right)$ and $g_{2} = \mathrm{exp}\left(\beta R_{2}/2\right)$ .

?

Problem 2.91 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

(i)(ii)(iii)

?

(i)

Let $t \in \mathbb {R}$ . Show that the matrix

g_{t} = \begin{pmatrix} \cosh t & i\sinh t \\ -i\sinh t & \cosh t \end{pmatrix}

has an inverse and find $g_{t}^{-1}$ . $g_{t}$ is an element of $SO(1,1)$ .

(ii)

The symmetric space $SL(2, \mathbb {C})/SU(2)$ can be identified with the space $H^{3}$ of positive $2 \times 2$ matrices with determinant $1$

A = \begin{pmatrix} a+b & x+iy \\ x-iy & a-b \end{pmatrix}, \qquad a,b,c,d \in \mathbb {R}

i.e. $\det A = a^{2}-b^{2}-x^{2}-y^{2} = 1$ . The group action is defined by

g_{t} \cdot A := g_{t}Ag_{t}^{*} .

Find $g_{t} \cdot A$ .

(iii)

Show that on the hyperboloid $a^{2} = 1+b^{2}+x^{2}+y^{2}$ , the matrix $g_{t}$ only changes the coordinates $y$ and $a$ .

Problem 2.93 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

(i)(ii)

We consider the Lie groups $U(2,2)$ , $SU(2,2)$ and the corresponding Lie algebras $u(2,2)$ , $su(2,2)$ , respectively. We define the matrix

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

Then

\begin{align} U(2,2) & := \left\{ G \in GL(4, \mathbb {C}) : G^{*}LG = L\right\} \\ SU(2,2) & := \left\{ G \in U(2,2) : \det G = 1\right\} \\ u(2,2) & := \left\{ X \in M_{4}(\mathbb {C}) : XL+LX = 0_{4}\right\} \\ su(2,2) & := \left\{ X \in u(2,2) : \tr (X) = 0\right\} \end{align}

where $M_{4}(\mathbb {C})$ is the vector space of the $4 \times 4$ matrices over the complex numbers.

?

(i)

Show that

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

is an element of $u(2,2)$ and $su(2,2)$ .

(ii)

Let $t \in \mathbb {R}$ . Calculate $\mathrm{exp}\left(tX\right)$ to find an element of $SU(2,2)$ .

Problem 2.102 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 2.1 "Programming Problems"

Let $\alpha \in \mathbb {R}$ . Given the matrix

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

Write a SymbolicC++ program that calculates the vector

A(\alpha ) \begin{pmatrix} x_{0} \\ x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}

and shows that $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{3}^{2}$ is invariant.

?

Elements of pseudo-orthogonal, pseudo-unitary and symplectic groups

References