Download 1 Warming up with rational points on the unit circle

4. Linear Algebra and Group Theory
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4.1 Quaternionic Geometry
Warming up with rational points on the unit circle
p p
A rational point on the unit circle takes the form q 1 , q22 on the Euclidean plane, satisfying that the pi , qi are
1
integers and also that
2 2
p2
p1
+
= 1.
q1
q2
The second condition here can be visualized geometrically by an application of the Pythagorean Theorem
(see the following diagram).
(0, 1)
•
p
1 p2
q1 , q2
•
p2
q2
p1
q1
We wish to find all of these points.
Imagine a line passing through the point (0, 1) with rational slope. It will meet the circle at two rational
points (one being (0, 1)). This now provides a technique for generating the rational points. That is to say,
each line through (0, 1) with rational slope will give rise to some rational point on the circle. Inversely, each
rational point gives rise to a line with rational slope through (0, 1).
So then, to find all the rational points, it will suffice to work with lines of the form
y = mx + 1
(1)
with m rational.
Calculating the point of intersection of the line parameterized by ( x, mx + 1) and the unit circle we must
have that
x2 + (mx + 1)2 = 1.
Rearranging here gives,
(1 + m2 ) x2 + 2mx = 0
which implies that x = 0 or x = 1−+2m
. The first solution corresponds to our fixed point (0, 1) and the second
m2
is the one of interest. Substituting into equation (2) gives
y=m
−2m
1 + m2
+1 =
1 − m2
.
1 + m2
1− m2
,
Hence, the rational points on the unit circle take the form 1−+2m
for each choice of rational m. With
m2 1+ m2
this, one can construct all of the Pythagorean triples. That is, write m = a/b for a and b integers, take not that
the triangle here has hypotenuse of length 1 so to scale the triangles by the lowest common denominator and
retrieve right angled triangles with only integer length sides.
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4. Linear Algebra and Group Theory
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4.1 Quaternionic Geometry
Arithmetic in the complex numbers
Introduction and motivation
Recall, we have the following nested chain of rings
Z⊂Q⊂R⊂C
and that the arithmetic (i.e. addition and multiplication) in each of these number systems becomes successively more intricate. In fact, Q, R and C are fields; meaning that every non-zero element admits a
multiplicative inverse and we have a notion of division! This is essentially why these number systems came
to be. For consider solving the equation 2x + 1 = 0 which suggest that x = − 12 . That is, one can no longer
find integer solutions to this equation and was lead to construct the rationals
Q=
p
: p, q ∈ Z
q
modulo equivalence by fractional simplification.
We shall assume a significant level of comfort in performing arithmetical operations in Q.
Now, in Z, we can add, subtract and multiply but have no sense of a multiplicative inverse. Allowing this
process brings us to working in Q which is still a discrete, countably infinite field. In searching for roots of
polynomials (for example, x2 − 2), we are forced to move into the real number, R, which can, in a different
light, be thought of analytically as the sequential completion of Q. Here, however, we are still missing roots
of polynomials such as x2 + 1 and the adjunction of such a root brings us up to complex numbers C. We will
review/define the properties of arithmetic in C, but it will be a good starting ground to recall the geometry
involved in complex multiplication.
Definition and basic notions in C
For the purpose of considering a number field containing the roots of x2 + 1, we define the complex numbers as
C := { a + bi : a, b ∈ R}
with component-wise addition given by
z1 + z2 = ( x1 + y1 i ) + ( x2 + y2 i ) = ( x1 + x2 ) + i ( y1 + y2 )
and having multiplication defined distributively with the convention that i2 = −1. In Euclidean coordinates
multiplication expands as follows;
z1 · z2 = ( x1 + iy1 ) · ( x2 + iy2 )
= x1 x2 + x1 y2 i + y1 x2 i + y2 x2 i 2
← this line is often omitted because its ugly
= ( x1 x2 − y1 y2 ) + i ( x1 y2 + x2 y1 )
where is it important to recognize, that in the second line we have allowed i to commute to the right hand
side1 .
1 this
will not necessarily be the case when we get to quaternions
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4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
In C, we have the following fundamental theorem about the existence of all roots to any polynomial with
complex coefficients. That is
Theorem 1 (Fundamental Theorem of Algebra). Any polynomial with complex coefficients has at least one complex
root.
Denote the real part of a complex number by
<(z) = <( a + bi ) = a =
z + z̄
2
where z̄ := a − bi denotes the complex conjugate of a complex number. In a similar fashion, denote the
imaginary part by
z − z̄
=(z) = =( a + bi ) = b =
2i
We have a notion of length or norm |z| of a complex number z by products as
|z|2 := z · z̄ = a2 + b2 ⇒ |z| =
p
a2 + b2
and this corresponds to the usual Pythagorean or Euclidean distance in the plane.
To see all of this geometrically, note that the complex numbers are isomorphic (as a vector space) to the
Euclidean plane R2 , having where the x and y axes represent the real and imaginary parts respectively.
Plotting a complex number here (as in the Euclidean plane) will reveal the norm formula as the standard
Euclidean distance of a vector from the origin.
Furthermore, C turns out to be field, so that every non-zero element has a multiplicative inverse. This is
given by
z̄
z −1 =
| z |2
All of these things will be important to keep in mind when we start working with quaternions.
Polar coordinates
As we have seen, there is an alternate representation for complex numbers, namely polar coordinates if we
represent a number on its unit circle as follows
eiθ := cos θ + i sin θ.
This definition agrees with all our prior knowledge of the basic trigonometric functions sine and cosine. Each
complex number z = a + bi in its Euclidean form corresponds uniquely to an element of the form r · eiθ .
Indeed,
z = a + bi = |z| ·
p
b
z
= a2 + b2 · ei arctan( a )
|z|
where here r = |z| and θ = arctan(b/a). All of this is summed up in the following diagram
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4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
reiθ
• eiθ
1
sin θ
θ
cos θ
Geometry of the multiplicative structure
Observe what happens when two complex numbers are multiplied together in polar coordinates: we have
that
z1 · z2 = r1 eiθ1 · r2 eiθ2 = (r1 · r2 )ei(θ1 +θ2 )
so that their radii are multiplied and the angles are added to each other. So, in fact, one can simply encode
rotations about the origin in the plane as multiplication by unit complex numbers. What can be said about
rotations about an arbitrary point?
Lemma. The rotation Rz,θ by an angle θ about z ∈ C is performed by translating back to the origin (by z) and
multiplying by eiθ and then translating back. That is, in terms of complex arithmetic,
Rz,θ : C → C
is defined as
Rz,θ (w) := eiθ (w − z) + z
In fact, this will allow us to complete the proof of the following
Propositon. The composition of two rotations by angles θ, ϕ is a rotation if θ + ϕ 6= 360◦ and a translation when
θ + ϕ = 360◦ (regardless of the centres of rotations).
Proof. Without loss of generality, we may assume that one of our rotations has the origin as its centre. So
beginning with rotations R p,θ and R0,ϕ . Now, as this may turn out different depending on the order in
which the rotations are applied, let us simply consider the composition R p,θ ◦ R0,ϕ (for the proof technique is
essentially identical).
We know that this composition is necessarily a rotation or translation as the resulting isometry is direct. Then,
in order to distinguish the two, we attempt to solve for a fixed point. That is, find z ∈ C such that
z = R p,θ ◦ R0,ϕ (z).
Expanding this says that
z = eiθ (eiϕ z − p) + p = ei(θ + ϕ) z + p(1 − eiθ ),
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4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
implying that
z=
p(1 − eiθ )
1 − ei (θ + ϕ)
which exists if and only if 1 − ei(θ + ϕ) 6= 0 and this is equivalent to the statement that θ + ϕ is not a multiple
of 360◦ .
Roots of unity
Now, we have already spoken about the factoring of any polynomial with complex coefficients and we can
further determine, from the fundamental theorem of algebra, that a polynomial of degree n has exactly
n-roots. In particular, let us solve for the roots of x n − 1.
It is easy to see that 1 is always a root here and this factors as x n − 1 = ( x − 1)( x n−1 + x n−2 + · · · + x + 1), or
expressed more cleanly
x n − 1 = ( x − 1) · Φ n ( x )
where
Φn ( x ) :=
n −1
∑
xi =
i =0
xn − 1
x−1
is used to denote the nth cyclotomic polynomial.
To find the remaining n − 1 roots here, suppose that z = reiθ is a root and substitute to see that r n einθ = 1.
This implies that r (being a positive real radius) is necessarily equal to 1 and that any value θ = 2πk
n for
n
k = 0, 1, 2, . . . , n − 1. Thus we have found all the roots of x − 1 and list them efficiently as
{1, ζ n , ζ n2 , . . . , ζ nn−1 }
where ζ n := e2πi/n is called a primitive nth root of unity.
Geometrically, we have found that the n roots of x n − 1 all lie on the unit circle in C and are evenly distributed
(equally spaced) by angles 2π/n. Pictorially, the roots of x4 − 1 and x5 − 1 are
ζ4 = i
ζ5
ζ 52
−1
2π
5
1
ζ 53
ζ 54
−i
so to notice that both ±1 are roots whenever n is even.
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4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
Using this theory, one achieves the following interesting result from Euclidean geometry
Proposition. Consider a regular n-gon inscribed in
the unit circle, fix your favourite vertex on the polygon and let a1 , a2 , . . . , an−1 represent the distances
from the fixed vertex to each of the other n − 1 vertices. The ai ’s are sketched in the following diagram
a2
a3
a4
Then
a1
n −1
a5
∏ a i = n2 .
i =1
Furthermore, if we let Fn denote the nth term in the Fibonacci sequence then also
n −1
∏ (5 − a2i ) = Fn2 .
i =1
Proof. Notice that each ai can be expressed as complex numbers by ai = |ζ ni − 1|. So then
n
∏ ai = ∏ |ζ ni − 1| = ∏(ζ ni − 1) = |Φn (1)| = n
i =1
i =1
i =1
n −1
n −1
For the second identity, we will use the same identification for the ai ’s but will also need to make use of the
following closed formula for the fibonacci sequence. That is
Fn =
where τ =
√
1+ 5
2
√
1− 5
2
and τ̂ =
τ n − τ̂ n
√
5
its “conjugate”. Expanding what we know,
n −1
n −1
n −1
n −1
i =1
i =1
i =1
i =1
∏ (5 − a2i ) = ∏ (5 − (ζ i − 1)(ζ −i − 1)) = ∏ (3 + ζ i + ζ −i ) = ∏ ζ n−i (3ζ i + ζ 2i + 1)
and the product ∏in=−11 ζ −i = ζ −
n ( n −1)
2
n −1
∏ (5 − a2i ) =
so that
∏in=−11 (ζ 2i + 3ζ i + 1)
i =1
ζ
n ( n −1)
2
= (−1)n−1
n −1
∏ (ζ 2i + 3ζ i + 1)
i =1
Now, we note that factoring the quadratic x2 + 3x + 1 via the quadratic formula, gives roots
−3 ±
√
32 − 4
2
Squaring our golden ratio
2
τ =
=
−3 ±
2
√
5
.
√ !2
√
√
1+ 5
6+2 5
3+ 5
=
=
2
4
2
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4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
and similarly,
√
3− 5
τ̂ =
.
2
2
So, in fact, we have that
x2 + 3x + 1 = ( x + τ 2 )( x + τ̂ 2 )
and the above expression becomes
n −1
n −1
i =1
i =1
n −1
∏ (5 − a2i ) = (−1)n−1 ∏ (ζ 2i + 3ζ i + 1)
= (−1)n−1
∏ (ζ i − τ2 )(ζ i − τ̂2 )
i =1
= (−1)n−1
(−τ 2 )n − 1 (−τ̂ 2 )n − 1
·
−τ 2 − 1
−τ̂ 2 − 1
= ···
(−1)n τ̂ 2n − 2 + (−1)n τ 2n
5
2n
n
2n
τ̂ − 2(−1) + τ
=
= Fn2 .
5
= (−1)n
Note that we have made use of the fact that τ τ̂ = −1.
There is a nice way of embedding all of this algebraic structure into the language of matrices (i.e. as a ring
homomorphism so to respect or preserve the additive and multiplicative structure of C). That is
Theorem 2. The map µ : C → M2 (R) given by
µ(z) := µ( a + bi ) =
a
−b
b
a
!
is an injective (field) homomorphism into the 2 × 2 real matrices.
Because of this, we have some cute little computational correspondences. For example, the real part , <(z),
carries over to the trace, the norm to the determinant, the inverse to the matrix inverse and so on. More
precisely,
|z|2 = zz̄ = a2 + b2 = det −ab ba = det(µ(z))
2<(z) = z + z̄ = 2a = Tr −ab ba = Tr(µ(z))
µ ( z −1 ) = µ ( z ) −1
Conjugation is encoded via the conjugation on M2 (R) as τµ(z)τ −1 where τ =
µ(z̄) = µ(z) T .
1 0
0 −1
. Alternatively
With all of this in mind, we can view the unit-complex numbers as the isomorphism of groups S1 ∼
= SO2 (R).
Furthermore, the purely imaginary complex numbers (a 1-dimensional real subspace of C when viewed as a
vector space) are isomorphic (as a vector space) to the 2 × 2 skew-symmetric real matrices, say Sym2− (R) and
the reals R sitting inside of C (as a subfield) are realized via µ by the 2 × 2 scalar matrices.
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4. Linear Algebra and Group Theory
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4.1 Quaternionic Geometry
Quaternions
Let H = { a + ib + jc + kd : a, b, c, d ∈ R} represent the skew-field of real quaternions. Addition here remains
coordinate-wise and multiplication is defined distributively with the following identifications
i2 = j2 = k2 = ijk = −1
I like to remember the multiplication by the following diagram,
i
j
k
whose arrows say that ij = k whereas ji = −k etc.
For a quaternion q = a + ib + jc + kd we define the norm , real part, purely imaginary part and conjugate of q in
the usual fashion as:
| q |2 : = a2 + b2 + c2 + d2 ,
<(q) := a,
ib + jc + kd and
q := a − ib − jc − kd,
respectively.
Furthermore, and again analogous to the C case, non-zero quaternions have inverses given by
q −1 =
q
| q |2
and a polar decomposition
q = |q| ·
q
|q|
which doesn’t look so special, until we see the isomorphism with a particular subset of M2 (C)
Again, there is a rotational multiplicative structure, with the exception that we have lost commutativity.
However this makes perfect sense because rotations didn’t commute in the first place in dimensions higher
than 2.
It is important to note here that, because we no longer have commutativity, the notation p/q does not make
sense. One must write pq−1 or q−1 p to distinguish between division on the right or left respectively.
3-dimensional geometry and H
If we restrict ourselves to purely imaginary quaternions (i.e. having real part equal to zero), then we are
essentially considering vectors in R3 .
The dot product can be expressed as
p·q =
1
1
( pq + qp) = (qp + pq),
2
2
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4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
the cross product as
p×q =
1
( pq − q̄ p̄).
2
More generally, if p = ps + ~pv , q = qs + ~qv are not restricted to being purely imaginary, but expressed in
terms of their scalar and vector components, then
pq = ps qs − ~pv · ~qv + ps~qv + qs~pv + ~pv × ~qv
C representation of quaternions
Notice, that a quaternion can be expressed as
q = a + ib + jc + kd = ( a + bi ) + (c + di ) j = c1 + c2 j,
where c1 = a + bi, c2 = c + di ∈ C. Here, the factor of j is must necessarily be placed on the right here
because of the fact that z · j = j · z̄ for z ∈ C following from the relation ij = − ji in H.
Now, as in the complex case, we have the following algebraic embedding of H into matrices.
!
c1 c2
Theorem 3. The map η : H → M2 (C) given by η (q) := η (c1 + c2 j) =
is an injective (division ring)
−c̄2 c̄1
homomorphism into the 2 × 2 complex matrices.
As above, this provides some linear algebraic expressions for inverses, conjugates, norms, etcetera. For
example,
|q|2 = det(η (q)),
1
<(q) = Tr η (q),
2
T
η (q) = η (q)∗ := η (q) ,
η ( q −1 ) = η ( q ) −1
What I would like to examine here is the (group of) unit length quaternions, (this is topologically a three
sphere) and the group isomorphism induced here from the linear algebraic perspective. Correspondingly
from the complex case we find the image of the unit quaternions to be 2 × 2 complex matrices of a very
particular form and having determinant 1. That is, inside M2 (C) the unit quaternions are precisely the family
{ A ∈ M2 (C) : A · A∗ = 1} otherwise known at the group of special unitary matrices and denoted SU2 (C). In
fact, this is a topologically smooth isomorphism of groups S3 ∼
= SU2 which furthermore smoothly preserves
the multiplicative and inversion structure of the groups.
Furthermore, combining these two theorems, we realize H as a subalgebra in the 4 × 4 real matrices as

µ ◦ η (q) = µ
c1
−c̄2
c2
c̄1
!
:=
µ ( c1 )
µ(−c̄2 )
µ ( c2 )
µ(c̄1 )
but whatever...
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!


=

a
-b
-c
-d
b
a
d
-c
c
-d
a
b
d
c
-b
a



 ∈ M4 (R)

4. Linear Algebra and Group Theory
4.1 Quaternionic Geometry
Hamilton
Sir William Rowan Hamilton (4 August 1805 2
September 1865) was an Irish physicist, astronomer,
and mathematician, who made important contributions to classical mechanics, optics, and algebra. His
studies of mechanical and optical systems led him to
discover new mathematical concepts and techniques.
His greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the
modern study of classical field theories such as electromagnetism, and to the development of quantum
mechanics. In mathematics, he is perhaps best known
as the inventor of quaternions.
Hamilton is said to have shown immense talent at a
very early age. In 1828, astronomer Bishop Dr. John
Brinkley remarked of the 18-year-old Hamilton, ’This
young man, I do not say will be, but is, the first mathematician of his age.’
1843. However, in 1840, Benjamin Olinde Rodrigues
had already reached a result that amounted to their
discovery in all but name.[6] Hamilton was looking
for ways of extending complex numbers (which can
be viewed as points on a 2-dimensional plane) to
higher spatial dimensions. He failed to find a useful 3-dimensional system (in modern terminology, he
failed to find a real, three dimensional skew-field), but
in working with four dimensions he created quaternions. According to Hamilton, on 16 October he was
out walking along the Royal Canal in Dublin with his
wife when the solution in the form of the equation
suddenly occurred to him; Hamilton then promptly
carved this equation using his penknife into the side
of the nearby Broom Bridge (which Hamilton called
Brougham Bridge), for fear he would forget it. This
event marks the discovery of the quaternion group.
William Rowan Hamilton’s scientific career included
the study of geometrical optics, classical mechanics,
adaptation of dynamic methods in optical systems,
applying quaternion and vector methods to problems
in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which
complex numbers are constructed as ordered pairs
of real numbers), solvability of polynomial equations
and general quintic polynomial solvable by radicals,
the analysis on Fluctuating Functions (and the ideas
from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the
space of quaternions (which is a special case of the
general theorem which today is known as the CayleyHamilton theorem). Hamilton also invented ”Icosian
Calculus”, which he used to investigate closed edge
paths on a dodecahedron that visit each vertex exactly
once.
The other great contribution Hamilton made to mathematical science was his discovery of quaternions in
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