Quaternion
A quaternion is a mathematical concept introduced by William Rowan Hamilton of Ireland in 1843. The idea captured the popular imagination for a time because it involves relatively simple calculations that abandon the commutative law, one of the basic rules of arithmetic. As such, it seemed to undermine one of the tenets of scientific knowledge.Specifically, a quaternion is a non-commutative extension of the complex numbers. As a vector space over the real numbers, the quaternions have dimension 4, whereas the complex numbers have dimension 2.
Table of contents |
1.1 Example
2 History1.2 Properties 1.3 Group rotation 1.4 Representing quaternions by matrices 1.5 Generalizations 3 See also 4 External links and resources |
Definition
· | 1 | i | j | k |
1 | 1 | i | j | k |
i | i | -1 | k | -j |
j | j | -k | -1 | i |
k | k | j | -i | -1 |
Every quaternion is a real linear combination of the unit quaternions 1, i, j, and k, i.e. every quaternion is uniquely expressible in the form a + bi + cj + dk. Addition of quaternions is accomplished by adding corresponding coefficients, as with the complex numbers. By linearity, multiplication of quaternions is completely determined by the multiplication table for the unit quaternions; this table is given at the right. Under this multiplication, the unit quaternions form the quaternion group of order 8, Q_{8}.
Example
Then
- x + y = 3 + 6i + j - 2k
- xy = (3 + i)(5i + j - 2k)
- = 15i + 3j - 6k + 5i^{2} + ij - 2ik
- = 15i + 3j - 6k - 5 + k + 2j
- = - 5 + 15i + 5j - 5k
Properties
Unlike real or complex numbers, multiplication of quaternions is not commutative: e.g. ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. The quaternions are an example of a division ring, an algebraic structure similar to a field except for commutativity of multiplication. In particular, multiplication is still associative and every non-zero element has a unique inverse.
Quaternions form a 4-dimensional associative algebra over the reals (in fact a division algebra) and contain the complex numbers, but they do not form an associative algebra over the complex numbers. The quaternions, along with the complex and real numbers, are the only finite-dimensional associative division algebras over the field of real numbers. The non-commutativity of multiplication has some unexpected consequences, among them that polynomial equations over the quaternions can have more distinct solutions than the degree of the polynomial.
The equation z^{2} + 1 = 0, for instance, has the infinitely many quaternion solutions z = bi + cj + dk with b^{2} + c^{2} + d^{2} = 1. The conjugate of the quaternion z = a + bi + cj + dk is defined as
By using the distance function d(z, w) = |z - w|, the quaternions form a metric space (isometric to the usual Euclidean metric on R^{4}) and the arithmetic operations are continuous. We also have |zw| = |z| |w| for all quaternions z and w. Using the absolute value as norm, the quaternions form a real Banach algebra.
Group rotation
As is explained in more detail in quaternions and spatial rotation, the multiplicative group of non-zero quaternions acts by conjugation on the copy of R^{3} consisting of quaternions with real part equal to zero. The conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(t) is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part. The advantages of Quaternions are:
- Non singular representation (compared with Euler angles for example)
- More compact (and faster) than matrices
- Pairs of unit quaternions can represent a rotation in 4d space.
Representing quaternions by matrices
There are at least two ways of representing quaternions as matrices, in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication (i.e., quaternion-matrix homomorphisms). One is to use 2×2 complex matrices, and the other is to use 4×4 real matrices.
In the first way, the quaternion a + bi + cj + dk is represented as
- All complex numbers (c = d = 0) correspond to matrices with only real entries.
- The square of the absolute value of a quaternion is the same as the determinant of the corresponding matrix.
- The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
- Restricted to unit quaternions, this representation provides the isomorphism between S^{3} and SU(2). The latter group is important in quantum mechanics when dealing with spin; see also Pauli matrices.
Generalizations
If F is any field and a and b are elements of F, one may define a four-dimensional unitary associative algebra over F by using two generators i and j and the relations i^{2} = a, j^{2} = b and ij = -ji. These algebras are either isomorphic to the algebra of 2×2 matrices over F, or they are division algebras over F. They are called quaternion algebras.
History
Quaternions were introduced by William Rowan Hamilton of Ireland in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. He could not do so for 3-dimensions, but 4-dimensions produce quaternions. According to the story Hamilton told, on October 16 Hamilton 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 into the side of the nearby Brougham Bridge (now called Broom Bridge). This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices were still in the future.
Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of real numbers, and described the first coordinate as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the dot product of the vector parts, while the vector part of the product is the cross product. But the significance of these was still to be discovered. Hamilton proceeded to popularize quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was published shortly after his death.
Use controversy
Even by this time there was controversy about the use of quaternions. Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (developed by Oliver Heaviside and Willard Gibbs among others), maintaining that quaternions provided a superior notation. While this is debatable in three dimensions, quaternions cannot be used in other dimensions (though extensions like octonions and Clifford algebras may be more applicable). Vector notation has nearly universally replaced quaternions in science and engineering by the mid-20th century.
James Clerk Maxwell described in the "A Dynamical Theory of the Electromagnetic Field" the interrelated nature of electricity, magnetism, and electromagnetic fields in a set of twenty differential equations in quaternions. The theory was the first paper in which Maxwell's equations appeared. Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, and, in 1873, he attempted a quaternion formulation. Quaterions have a vector and a scalar part and have a higher topology than vector and tensor analysis. The theory unifies two kinds of force - the electric and the magnetic.
Oliver Heaviside reduced the complexity of Maxwell's quaternion equations, creating the four vector-based differential equations we now know collectively as Maxwell's equations. Some people claim that Maxwell's original quaternion equations describe certain physical effects that cannot be explained by the simplified vector equations.
Recent years
Quaternions are sometimes used in computer graphics (and associated geometric analysis) to represent rotations or orientations of objects in 3d space. Quaternions also see use in control theory, signal processing, attitude control, physics, and orbital mechanics, mainly for representing rotations/orientations in three dimensions. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations.
Since 1989, the National University of Ireland, Maynooth has organized a pilgrimage, where mathematicians (including Murray Gell-Mann in 2002 and Andrew Wiles in 2003) take a walk from Dunsink observatory to the Royal Canal bridge where, unfortunately no trace of the Hamilton's carving remains.
See also
- Complex number
- Octonion
- Hypercomplex number
- Division algebra
- Associative algebra
- Quarternion by Sofia Gubaidulina
External links and resources
- Doing Physics with Quaternions
- Quaternion Calculator [Java]
- The Physical Heritage of Sir W. R. Hamilton (PDF)
- Kuipers, Jack (2002). Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality (Reprint edition). Princeton University Press. ISBN 0691102988
- "Quaternion". 1911 encyclopedia.
- Tait, Peter Guthrie, "Quaternion". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160-164. (PostScript file)
Topics in mathematics related to quantity |
Numbers | Natural numbers | Integers | Rational numbers | Real numbers | Complex numbers | Hypercomplex numbers | Quaternions | Octonions | Sedenions | Hyperreal numbers | Surreal numbers | Ordinal numbers | Cardinal numbers | p-adic numberss | Integer sequences |Mathematical constants | Infinity |