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Transcript
```Hamilton’s Quaternions
by Michele Laino
August 4, 2015
Abstract
In this tutorial I will show the Hamilton’s quaternions and their main properties, after
the necessary definitions, I will expose the algebra of such mathematical objects.
Let H the set of the subsequent matrices of M2 (C):
z
w
q=
−w̄ z̄
where z, w ∈ C, and, as usual, z̄ is the complex conjugate of z.
As we can note the determinant of one of such matrix above, is:
z
w
2
2
det (q) = det
= z z̄ + ww̄ = |z| + |w|
−w̄ z̄
which is equal to 0 if and only if z = w = 0.
The non-negative square root of det (q) is called the modulus of q, and is indicated as |q|:
2
|q| = det (q) ,
q∈H.
If (z, w) 6= (0, 0), the inverse of that above matrix, is:
z
−w̄
w
z̄
−1
=
1
2
2
|z| + |w|
z̄
w̄
−w
z
which is again a matrix of H. Furthermore, given the subsequent four complex
numbers z, w, u, v the product:
z
w
u v
zu − wv̄
zv + wū
=
=
−w̄ z̄
−v̄ ū
−w̄u − z̄ v̄ −w̄v + z̄ ū
=
zu − wv̄
zv + wū
−(zv + wū) (zu − wv̄)
1
is again an element of H.
The set H of the not null matrices is then a group with respect to the operation
of product rows times columns. Of course H is an abelian group with respect to
the sum of matrices. The sum and the product rows times columns of matrices,
have the distributive property. So we conclude that with those operations as
above defined H is a field : it is called the field of quaternions of Hamilton. The
field of complex numbers is identified to a sub-field of the field of the quaternions
of Hamilton, through the subsequent embedding:
z 0
z∈C→
∈H
0 z̄
If we denote with j, the subsequent matrix:
0 1
j=
−1 0
and we identify the complex numbers to the corresponding matrix of H, then a
quaternion can be rewritten, in a unique way, as follows:
z
−w̄
w
z̄
=
z
0
0
z̄
+
w
0
0
w̄
0
·
−1
1
0
= z + wj .
Theorem 1 The operations of addition and product between quaternions, can
be rewritten as follows:
(z + wj) + (u + vj) = (z + u) + (w + v) j ,
(z + wj) · (u + vj) = (zu − wv̄) + (zv + wū) j ,
∀z, w, u, v ∈ C .
Proof.
(z + wj) + (u + vj) =
=
=
z
0
0
z̄
+
0
−w̄
z+u
w+v
− (w̄ + v̄) z̄ + v̄
w
0
+
u 0
0 ū
+
= (z + u) + (w + v) j
2
0
−v̄
v
0
=
(z + wj) · (u + vj) =
=
z
−w̄
w
z̄
u
−v̄
v
ū
zu − wv̄
−w̄u − z̄ v̄
zv + wū
−w̄v + z̄ ū
0
−w̄u − z̄ v̄
zv + wū
0
=
(zu − wv̄) + (zv + wū) j =
=
=
zu − wv̄
0
zu − wv̄
−w̄u − z̄ v̄
0
−w̄v + z̄ ū
zv + wū
−w̄v + z̄ ū
+
=
Writing a complex number as element of H, we can note this:
zj = jz̄
j2 = j · j = −I2
where I2 , is the identity matrix.
Next, we introduce the subsequent other matrices:
i 0
0
1 = I2 , i =
, k=
0 −i
i
i
0
being, as usually, i2 = −1.
Using, the matrices 1, i, j, k, we can write every quaternion q, in a unique way,
as below:
.
q = x0 1 + x1 i + x2 j + x3 k =
x0 + ix1
−x2 + ix3
x2 + ix3
x0 − ix1
,
x0 , x1 , x2 , x3 ∈ R
The field R of the real numbers is identified to a sub-field of the field H of
quaternions, by means of this topological embedding:
x ∈ R → x1 ∈ H .
The expression of quaternions as linear combination of the matrices 1, i, j, k
highlights that the field of quaternions is an extension of the real number’s field.
Here are some computation rules:
i2 = j2 = k2 = −1,
ij = −ji = k,
3
jk = −kj = i,
ki = −ik = j
On the field H is defined an anti-involution, which is called conjunctive transformation, and it is given by this definition:
z + wj = z̄ − wj
∀z, w ∈ C
and hence:
q̄ = x0 1 + x1 i + x2 j + x3 k = x0 1 − x1 i − x2 j − x3 k
∀x0 , x1 , x2 , x3 ∈ R .
Theorem 2 The conjunctive transformation satisfies the subsequent properties:
q1 q2 = q̄2 q̄1
q1 + q2 = q̄1 + q̄2
2
qq̄ = |q|
q + q̄ = 2<q ∈ R
∀q, q1 , q2 ∈ H
Proof.
a)
q1 q2 = (z1 + w1 j) · (z2 + w2 j) =
= (z1 z2 − w1 w̄2 ) + (z1 w2 + w1 z̄2 ) j ⇒
⇒ q1 q2 = (z̄1 z̄2 − w̄1 w2 ) − (z1 w2 + w1 z̄2 ) j
q̄2 q̄1 = (z2 + w2 j) · (z1 + w1 j) =
= (z̄2 z̄1 − w2 w̄1 ) + (−z̄2 w1 − w2 z1 ) j
b)
q1 + q2 = (z1 + w1 j) + (z2 + w2 j) =
= (z1 + z2 ) + (w1 + w2 ) j ⇒
⇒ q1 + q2 = (z̄1 + z̄2 ) − (w1 + w2 ) j,
q̄1 + q̄2 = (z1 − w1 j) + (z2 − w2 j) =
= (z1 + z2 ) − (w1 + w2 ) j
c)
2
2
qq̄ = (z + wj) · (z̄ − wj) = |z| + |w| + (−zw + wz) j =
2
2
= |z| + |w|
4
d)
q + q̄ = (z + wj) + (z̄ − wj) = z + z̄ .
From the third property of the theorem above, with z = x0 + ix1 , w = x2 + ix3 ,
we can write:
qq̄ = x20 + x21 + x22 + x23 .
The non-negative real number:
|q| =
√
qq̄ =
q
x20 + x21 + x22 + x23
is called the modulus of the quaternion q.
Theorem 3 The modulus of a quaternion has the subsequent properties:
|q|
is a real number
|q| = 0
if and only if q = 0
|q| = |q̄|
|q1 q2 | = |q1 | |q2 |
|q1 + q2 | 6 |q1 | + |q2 |
and the equality holds if and only if one of the two
quaternions is a real, and non − negative multiple of the other one.
Proof. The first four properties are easy to prove, whereas for the last one, we
have:
2
|q1 + q2 | =
2
2
2
2
2
2
2
2
= |q1 | + |q2 | + q1 q̄2 + q2 q̄1
= |q1 | + |q2 | + 2< (q1 q̄2 )
6 |q1 | + |q2 | + 2 |q1 q̄2 |
6 |q1 | + |q2 | + 2 |q1 | |q2 |
2
6 (q1 + q2 ) .
In order for the equality is checked, it is necessary and sufficient that:
q1 q̄2 + q2 q̄1 = 2 |q1 | |q2 |
namely, q1 q̄2 is a positive real number, say r. So, if q1 q̄2 6= 0, then we can
multiply the subsequent equality:
q1 q̄2 = r
from right, by q2 , so we get:
5
q1 q̄2 q2 = rq2
2
q1 |q2 | = rq2
and hence, the two quaternions are one multiple of the other one, with respect
to a real and positive number.
The case in which one of the two quaternions is null is trivial.
*
*
6
*
```