UCLA Math Circle: Quaternions
10/20/2013
• Motivations and definitions:
A complex numbers a + bi can be considered as a pair of real numbers (a, b). One
could add and multiply pairs like (a, b) and (c, d) such that the distributive property
also holds. This is then known as an algebra over R. Furthermore, the operations are
commutative, associative and multiplicative inverses exist for every nonzero complex
number. So one could define division as multiplying the multiplicative inverse. Set of
numbers satisfying these properties is call a commutative, division algebra.
Definition 1. The set Rn with a multiplication
m : Rn × Rn −→ Rn
(x, y) 7→ m(x, y)
is said to be an R-algebra, or real algebra, if the multiplication the distribution laws
holds, i.e.
m((αx + βy), z) = αm(x, z) + βm(y, z),
m(x, (αy + βz)) = αm(x, y) + βm(x, z),
for any α, β ∈ R, x, y, z ∈ Rn .
Definition 2. An algebra A is a division algebra if given a, b ∈ A with a 6= 0, the two
equations aX = b and Xa = b have unique solutions in A.
Example:
1. R and C are both real division algebras.
2. Mat(2, R), the set of 2 × 2 matrices with real entries is an R-algebra, but not
a division algebra. Similarly, Mat(2, C), the set of 2 × 2 matrices with complex
entries is an R-algebra.
• A Model of the Quaternions
For R4 , we could define a multiplication as follows. Let
e = e1
i = e2
j = e3
k = e4
= (1, 0, 0, 0),
= (0, 1, 0, 0),
= (0, 0, 1, 0),
= (0, 0, 0, 1).
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Then every element in R4 can be written as αe + βi + γj + δk for unique α, β, γ, δ ∈ R.
Define the multiplication by
i2 = j 2 = k 2 = ijk = −e.
(1)
The set R4 with this operation is called the quaternions, denoted by H.
Exercise: Derive the following commutation relationships from equation (1) above.
ij = −ji,
jk = −kj,
ki = −ik.
Exercise: Find the product
(αe + βi + γj + δk)(α0 e + β 0 i + γ 0 j + δ 0 k).
Exercise: Verify that for any x = αe + βi + γj + δk, we have
x2 = 2αx − (α2 + β 2 + γ 2 + δ 2 )e.
• Another Model of H
There is a map between R-algebras
Φ : C −→ Mat(2, R),
a −b
a + bi 7→
,
b a
The image of Φ is a subalgebra of Mat(2, R). The following exercise shows that the
subalgebra of Mat(2, R) behaves the same way as C. In this case, we say that the map
is a homomorphism.
Exercise: Verify that Φ satisfies the following properties
1. Φ(1) = ( 10 01 ) .
2. cΦ(a + bi) = Φ(c(a + bi)).
3. Φ(a + bi) + Φ(c + di) = Φ((a + bi) + (c + di)).
4. Φ(a + bi)Φ(c + di) = Φ((a + bi)(c + di)).
2
Similarly, there is a homomorphism between H and a subalgebra of Mat(2, C). Let
1 0
i 0
0 −1
0 −i
E=
,I =
,J =
,K =
.
0 1
0 −i
1 0
−i 0
The map is described as follows.
Φ : H −→ Mat(2, C),
α + βi −γ − δi
(α, β, γ, δ) 7→
= αE + βI + γJ + δK.
γ − δi α − βi
(2)
As we saw earlier, it is easier to compute with quaternions than with 2 × 2 matrices.
The set {E, −E, I, −I, J, −J, K, −K} is a group under multiplication. It is the (finite)
quaternion group. Cayley was familiar with representing quaternions using 2 by 2
matrices (1858).
Let SU(2) = {A ∈ Mat(2, C)|AĀt = E, det(A) = 1}. Then we have
w −z
2
2
SU(2) =
∈ Mat(2, C) : |w| + |z| = 1 .
z̄ w̄
There is an isomorphism between SU(2) and the unit quaternions from the algebra
homomorphism (2) above.
• H as a Euclidean Space
Let x = αe + βi + γj + δk, y similarly defined. Then we can define conjugate of x to
be
x̄ = αe − βi − γj − δk.
(3)
With respect to multiplication, we have
xy = ȳx̄.
(4)
Then we can define the magnitude of x ∈ H to be
|x|2 = xx̄.
If a quaternion has magnitude 1, then we call it a unit quaternion. Notice we have the
“four square theorem”:
|wz|2 = (wz)(wz) = wz z̄ w̄ = w|z|2 w̄ = |w|2 |z|2 .
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Similar to the complex numbers, let
Re(x) = α, Im(x) = βi + γj + δk.
From that, we have
x − x̄
x + x̄
, Imx =
.
2
2
Then we can define a scalar product on H as
Re(x) =
(5)
hx, yi := αα0 + ββ 0 + γγ 0 + δδ 0 .
(6)
Exercise:
1. Find |i|2 , hi, ji, hi, ki etc.
2. Prove the orthogonality criterion:
hx, yi = 0 ⇔ xȳ = −yx̄.
3. Show that the inner product can be expressed in terms of conjugates,
hx, yi = Re(xȳ) =
xȳ + yx̄
.
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4. Prove that hx, yi is symmetric and R-linear
hx, yi = hy, xi.
5. Verify that
|x|2 = xx̄ = Re(xx̄) = hx, xi = α2 + β 2 + γ 2 + δ 2 .
• Imaginary space of H
The imaginary space of H is defined to be
ImH := {bi + cj + dk : b, c, d ∈ R}.
By previous exercise, x ∈ ImH if and only if x2 < 0.
Exercise: Show that for purely imaginary quaternions u, v, we have uv + vu ∈ Re.
Given purely imaginary quaternions u, v, then we have
uv = −hu, vie + u × v,
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(7)
where u × v is the vector cross product. From this and u × v = −v × u, we have
1
1
u × v = (uv − vu), hu, vie = − (uv + vu)∀u, v ∈ ImH.
2
2
Exercise: Show that every nonzero quaternion x ∈ H can be represented in infinitely
many different ways as the product x = uv of two purely imaginary quaternions u, v
Exercises: Derive the following formula
1. u × (v × w) = 12 (uvw − vwu).
2. (Grassman identity): u × (v × w) = hu, wiv − hu, viw.
3. (Jacobi identity): u × (v × w) + v × (w × u) + w × (u × v) = 0.
By taking the magnitudes of both sides of equation (7), we have
hu, vi2 + |u × v|2 = |u|2 |v|2 .
If we write hu, vi = |u||v| cos ϕ, then |u × v| = |u||v| sin ϕ.
• Spatial Rotation The imaginary quaternions can be identified with R3 and the
quaternions are closely related to the rotation of three space.
Theorem Let q = a + bi + cj + dk be a unit quaternion. We can write it as
q = cos(α/2) + u sin(α/2),
where u ∈ ImH and has magnitude 1. Then the following map is the rotation of R3
with respect to the axis u by a degree of α in the counterclockwise direction
f : R3 → R3 , v 7→ qvq −1 .
Proof. Write q = cos(α/2) + u sin(α/2). Then
qvq −1 =
=
=
=
=
=
(cos(α/2) + u sin(α/2))v(cos(α/2) − u sin(α/2))
cos2 (α/2)v + sin(α/2) cos(α/2)(uv − vu) − sin2 (α/2)uvu
cos(α)v + sin2 (α/2)v + sin(α)u × v − sin2 (α/2)uvu
cos(α)v + sin(α)u × v + sin2 (α/2)(v − uvu)
cos(α)v + sin(α)u × v + sin2 (α/2)2hu, viu
cos(α)v + sin(α)u × v + (1 − cos(α))hu, viu.
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Now if hv, ui = 0 and hv, vi = 1, then
hqvq −1 , vi = cos(α)hv, vi + sin(α)h(u × v), vi + 0
= cos(α).
Similarly,
hqvq −1 , u × vi = sin(α).
So qvq −1 is a rotation around axis u with angle α in the counterclockwise direction.
Corollary: Compositions of rotations are still rotations.
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