The Lumenary · Advanced undergraduate

Matrix groups and continuous symmetry

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You will learn to

Define the classical matrix groups $GL_n$ , $SL_n$ , $O(n)$ , $SO(n)$ , $U(n)$ , $SU(n)$ and the Euclidean group, and decide which are compact and which are connected
Prove Euler's rotation theorem and that $SO(3)$ , unlike $SO(2)$ , is not commutative
Use unit quaternions to describe rotations of $\R^3$ and construct the two-to-one homomorphism $SU(2) \to SO(3)$ with kernel $\{\pm I\}$
Compute matrix exponentials, prove that $t \mapsto \exp(tX)$ is a one-parameter subgroup with tangent vector $X$ , and prove $\det(\exp X) = e^{\operatorname{tr} X}$
Identify the Lie algebra of a matrix group with its tangent space at the identity, show that the Lie algebra of $SO(n)$ is the skew-symmetric matrices, and compute brackets

1.
Which of these real $3 \times 3$ matrices lie in $SO(3)$ ? (Choose all that apply.)
- A. $\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$
- B. $-I$
- C. $\operatorname{diag}(1, 1, -1)$
- D. $\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
- E. $2I$
- F. $\operatorname{diag}(-1, -1, 1)$
2.
A rotation $A \in SO(3)$ has $\operatorname{tr} A = 0$ . Through what angle $\theta \in [0°, 180°]$ does it rotate, in degrees?
3.
What is the dimension of the rotation group $SO(5)$ (equivalently of its Lie algebra $\mathfrak{so}(5)$ )?
4.
Which of these matrix groups are compact? (Choose all that apply.)
- A. $O(3)$
- B. $SL_2(\mathbb{R})$
- C. $U(2)$
- D. $GL_2(\mathbb{R})$
- E.The Euclidean group $E(2)$
- F. $SU(2)$
5.
The proof of Euler's theorem shows $\det(A - I) = -\det(A - I)$ for $A \in SO(3)$ . Which step uses that the dimension is odd?
- A. $\det(I - A) = \det(-(A - I)) = (-1)^3 \det(A - I)$
- B. $A - I = A(I - A^{\mathsf T})$
- C. $\det(A(I - A^{\mathsf T})) = \det(I - A^{\mathsf T})$
- D. $\det(I - A^{\mathsf T}) = \det(I - A)$
6.
The unit quaternion $q_\theta = \cos\frac\theta2 + \sin\frac\theta2\,k$ induces the rotation $\rho(q_\theta)$ by $\theta$ about the $z$ -axis. As $\theta$ increases from $0°$ to $360°$ , the rotation returns to the identity. What is the real part of $q_\theta$ at $\theta = 360°$ ?
7.
The Lean theorem exp_skew_orthogonal concludes (NormedSpace.exp X)ᵀ * NormedSpace.exp X = 1 for a real square matrix X. Which hypothesis makes it true?
- A.hX : Xᵀ = -X
- B.hX : Xᵀ = X
- C.hX : X.trace = 0
- D.hX : X * X = 0
8.
For which pairs is $\exp(X + Y) = \exp X \exp Y$ ? Here $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ , $N' = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ , $J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ , $E = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ . (Choose all that apply.)
- A. $X = \operatorname{diag}(1, 2)$ , $Y = \operatorname{diag}(3, 4)$
- B. $X = N$ , $Y = N'$
- C. $X = J$ , $Y = 2J$
- D. $X = N$ , $Y = I$
- E. $X = E$ , $Y = N$
9.
Implement Hamilton's product qmul(p, q) for quaternions given as tuples (a, b, c, d) meaning $a + bi + cj + dk$ , and rotate(q, v) returning the vector part of $q\,v\,\bar q$ for a unit quaternion q and a vector v = (x, y, z) (identified with the pure quaternion $xi + yj + zk$ ).
Computational — Practice

Answer key — Matrix groups and continuous symmetry

1.
In $SO(3)$ : the axis-cycling permutation matrix (an even permutation, determinant $1$ ), the quarter turn about $z$ , and $\operatorname{diag}(-1, -1, 1)$ , the half turn about $z$ . Not in $SO(3)$ : $-I$ and $\operatorname{diag}(1, 1, -1)$ (orthogonal with determinant $-1$ ) and $2I$ (not orthogonal).
2.
$1 + 2\cos\theta = 0$ gives $\cos\theta = -\tfrac12$ , so $\theta = 120°$ . The axis-cycling permutation matrix of the previous exercise has trace $0$ and is indeed the $120°$ turn about $(1, 1, 1)$ .
3.
$\dim \mathfrak{so}(5) = 5 \cdot 4/2 = 10$ . Check with $n = 3$ : $\dim SO(3) = 3$ (an axis and an angle), and $n = 2$ : $\dim SO(2) = 1$ (an angle).
4.
Compact: $O(3)$ , $U(2)$ , $SU(2) \cong S^3$ — closed and bounded because their columns are unit vectors. Not compact: $SL_2(\mathbb{R})$ ( $\operatorname{diag}(t, 1/t)$ ), $GL_2(\mathbb{R})$ , and $E(2)$ (translations).
5.
Only $\det(-M) = (-1)^n\det M$ depends on $n$ . For $n = 3$ it gives $\det(A - I) = -\det(A - I)$ , so $\det(A - I) = 0$ . For even $n$ the argument gives nothing, and indeed $R(90°) \in SO(2)$ and $\operatorname{diag}(R(\theta), R(\varphi)) \in SO(4)$ (with $\theta, \varphi \ne 0$ ) have no fixed vectors.
6.
$q_{360°} = \cos 180° + \sin 180°\, k = -1$ . The rotation $\rho(-1) = \rho(1) = I$ , but the path of quaternions went from $1$ to $-1$ , half of a great circle in $S^3$ ; it closes up only at $\theta = 720°$ . This is the sign change of a spin- $\tfrac12$ state under a full turn.
7.
The hypothesis is skew-symmetry Xᵀ = -X, i.e. X ∈ 𝔰𝔬(n). The proof is rw [← Matrix.exp_transpose, hX, Matrix.exp_neg] followed by A⁻¹ * A = 1 for the invertible matrix A = exp X. The converse direction — every tangent vector to O(n) at I is skew — is the separate theorem tangent_skew.
8.
Commuting pairs: the diagonal pair, $J$ and $2J$ , and $N$ with $I$ . The pairs $(N, N')$ and $(E, N)$ do not commute and the identity fails, as computed. Commuting is sufficient but not necessary: there are non-commuting $X, Y$ with $\exp(X + Y) = \exp X \exp Y$ (for instance with eigenvalues in $2\pi i \mathbb{Z}$ ), which is why the safe general statement is Theorem 7(a) plus the BCH formula.
9.
```
def qmul(p, q):
    a1, b1, c1, d1 = p
    a2, b2, c2, d2 = q
    return (a1 * a2 - b1 * b2 - c1 * c2 - d1 * d2,
            a1 * b2 + b1 * a2 + c1 * d2 - d1 * c2,
            a1 * c2 - b1 * d2 + c1 * a2 + d1 * b2,
            a1 * d2 + b1 * c2 - c1 * b2 + d1 * a2)


def rotate(q, v):
    a, b, c, d = q
    w = qmul(qmul(q, (0,) + tuple(v)), (a, -b, -c, -d))
    return tuple(w[1:])
```
The $i$ -coefficient of the product, for instance, collects $a_1 b_2 + b_1 a_2$ (real times $i$ ) and $c_1 d_2 - d_1 c_2$ (from $jk = i$ and $kj = -i$ ). With $q = \tfrac12(1 + i + j + k)$ the rotation is the $120°$ turn about $(1, 1, 1)$ , sending $x \to y \to z \to x$ .