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4. Linear Algebra and Group Theory 1 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. 1 4. Linear Algebra and Group Theory 2 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 2 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 3 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 ), 4 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. 5 1 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 6 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. 7 4. Linear Algebra and Group Theory 3 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 8 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... 9 ! = 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 10