Important examples from linear algebra

The groups one studies in geometry are usually groups of matrices. The simplest example is the group of $1 \times 1$ inversible matrices. This is just the set of nonzero real numbers with their standard multiplication. The higher dimensional analogue is the group of $n \times n$ inversible matrices provided with their usual multiplication.

Before you go on, make sure you can effortlessly solve the following exercises:

Exercise 1.4   Multiply the matrices

$$\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$    and $$\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}^{-1}$$

Exercise 1.5   Mentally compute the inverse of the matrix

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

Exercise 1.6   Compute the determinant of

$$\begin{pmatrix} 1 & 1 & 0 \\ 5 & 1 & 1 \\ 2 & 0 & 1 \end{pmatrix}^{-5}$$

If you cannot solve these problems in a very short period of time, then you are not ready for this course. Contact your instructor.

Besides the group $GL(n,{\Bbb R})$ of invertible $n \times n$ matrices of real numbers, we also have the group $GL(n,{\Bbb C})$ of invertible $n \times n$ matrices of complex numbers. Notice that $GL(n,{\Bbb R})$ is a subgroup of $GL(n,{\Bbb C})$. In fact, almost every interesting group in geometry is a subgroup of $GL(n,{\Bbb C})$. Here are some examples:

The special linear group. The set of real (resp. complex) $n \times n$ matrices having determinant one is denoted by $SL(n, {\Bbb R})$ (resp. $SL(n,{\Bbb C})$) and is called the special linear group. If $A \in SL(n,{\Bbb R})$, then the transformation from ${\Bbb R}^n$ to itself defined by $v \mapsto A v$ preserves volume and orientation.

The orthogonal and the special orthogonal group. A square matrix is said to be orthogonal if its transpose is equal to its inverse. The set of $n \times n$ orthogonal matrices is denoted by $O(n)$. The subset of $O(n)$ consisting of matrices of determinant one is denoted by $SO(n)$. They are respectively the orthogonal and the special orthogonal group.

Exercise 1.7 (05)   Let us denote the scalar or inner product of two vectors $v$ and $w$ in ${\Bbb R}^n$ by $v \cdot w$.

1. Show that if $A$ is any $n \times n$ matrix and $A^{t}$ is its transpose, then $Av \cdot w = v \cdot A^{t}w$. Conclude that if $A \in O(n)$, then $Av \cdot Aw = v \cdot w$.
2. Show that if $A$ is an orthogonal matrix, the distance between $v$ and $w$ equals the distance between $Av$ and $Aw$.
3. Show that the determinant of an orthogonal matrix is either $1$ or $-1$

The unitary and the special unitary group. If $A$ is a square matrices of complex numbers, the adjoint of $A$, denoted by $A^{*}$, is the result of transposing $A$ and then conjugating all of its entries. For example

$$\begin{pmatrix} 1+ i & 0 \\ i & 1 \end{pmatrix}^{*} = \begin{pmatrix} 1- i & -i \\ 0 & 1 \end{pmatrix}$$

A square complex matrix is said to be unitary if its adjoint is equal to its inverse. The set of $n \times n$ unitary matrices is denoted by $U(n)$. The subset of $U(n)$ consisting of matrices of determinant one is denoted by $SU(n)$. They are respectively the unitary and the special unitary group.

Exercise 1.8 (05)   Let us define the Hermitian product of two vectors $v := (v_{1},\dots v_{n})$ and $w := (w_{1},\dots,w_{n})$ in ${\Bbb C}^n$ by $<v,w> := v_{1}\bar{w}_{1} + \cdots v_{n}\bar{w}_{n}$.

1. Show that if $A$ is any $n \times n$ complex matrix and $A^{*}$ is its adjoint, then $<Av,w> = <v,A^{*}w>$. Conclude that if $A$ is unitary, then
   $$<Av,Aw> = <v,w>.$$

2. Show that if $A$ is a unitary matrix, the distance between $v$ and $w$ equals the distance between $Av$ and $Aw$.
3. Show that the determinant of an unitary matrix is a complex number of modulus one.

Exercise 1.9 (*10)   In what follows $I_n$ will denote the $n \times n$ identity matrix and $J_{2n}$ is the $2n \times 2n$ matrix $\begin{pmatrix} 0 & I_{n} \\ -I_{n} & 0 \end{pmatrix}$.

1. Show that the set of all $2 \times 2$ real matrices that commute with $J_{2}$ can be naturally identified with the set of complex numbers.
2. By analogy with the previous item, show that the group $GL(n,{\Bbb C})$ can be seen as a subgroup of $GL(2n, {\Bbb R})$.
3. Define the symplectic product of two vectors on ${\Bbb R}^{2n}$ by $\omega(v,w) := v^{t}Jw$. Show that the symplectic product is antisymmetric and nondegenerate.
4. Define the linear symplectic group $Sp(2n)$ as the set of all $2n \times 2n$ real matrices $A$ satisfying $A^{t}JA = J$. Show that the intersection of $Sp(2n)$ and $GL(n,{\Bbb C})$ equals $U(n)$.

Exercise 1.10 (15)   Show that the group $SU(2)$ can be identified with the $3$-dimensional unit sphere in ${\Bbb C}^2 = {\Bbb R}^4$.

Definition 1.3   The trace of a square matrix is the sum of all the elements in its diagonal. The commutator of two $n \times n$ matrices $X$ and $Y$, denoted by $[X,Y]$, is defined as $XY - YX$

Exercise 1.11 (05)   In what follows $A$, $X$, and $Y$ are $n \times n$ matrices with $A$ invertible

1. Show that $[AXA^{-1}, AYA^{-1}] = A[X,Y]A^{-1}$.
2. Show that if $X$ has zero trace, then $AXA^{-1}$ has zero trace.

The following exercise was taken from the book of Abraham and Marsden on the foundations of mechanics (see [1]).

Exercise 1.12 (15)   Let $su(n)$ be the set of $n \times n$ complex matrices $X$ of zero trace such that $X^* = -X$.

1. Show that $su(n)$ is a vector space and compute its dimension.
2. Show that if $A \in SU(n)$ and $X \in su(n)$, then $AXA^{-1} \in su(n)$.
3. The three Pauly spin matrices from quantum mechanics are
   $$\sigma_{1} := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_{2} := \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},$$    and $$\sigma_{3} := \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$
   Show that the matrices $\gamma_{i} = (i/\sqrt{2})\sigma_{i}, i = 1,2,3$, form a basis of $su(2)$ with the commutation relations
   $$[\gamma_{i},\gamma_{j}] = \epsilon_{ijk}\gamma_{k} ,$$
   where $\epsilon_{ijk}$ equals $1$ if $ijk$ is an even permutation of $1 2 3$ and $-1$ otherwise.

4. Indentify ${\Bbb R}^3$ with $su(2)$ by assigning to each vector $x = (x_{1}, x_{2}, x_{3})$ the matrix
   $$x \cdot \gamma := x_{1}\gamma_{1} + x_{2}\gamma_{2} + x_{3}\gamma_{3}.$$
   If $x \wedge y$ denotes the vector product of two vectors in ${\Bbb R}^3$, show that
   $$[(x \cdot \gamma),(y \cdot \gamma)] = (x \wedge y) \cdot \gamma .$$

5. Show that the determinant of $(x \cdot \gamma)$ equals ${1\over 2} \Vert x\Vert^{2}$.
6. Let $A$ be in $SU(2)$ and consider the transformation $T_{A}$ from ${\Bbb R}^3$ to itself defined by $A(x \cdot \gamma)A^{-1} = T_{A}x \cdot \gamma$. Show that $T_{A}$ is a linear transformation that preserves distances and has determinant one.
7. Show that if $A$ and $A'$ are in $SU(2)$, then $T_{A} = T_{A'}$ if and only if $A = \pm A'$.

From this exercise we conclude that the group $SU(2)$, which we saw earlier identified with the $3$-dimensional sphere, has a natural $2$-to-$1$ correspondence with the group of rotations $SO(3)$. To this quirk of nature we owe the spin of electrons and a pretty little trick due to Dirac and which we shall describe in chapter 5.