Quaternion

Quaternions are an extension of the real numbers, similar to the complex numbers. While the real numbers are extended to the complex numbers by adding a number i such that i² = -1, quaternions are extended by adding elements i, j and k to the real numbers such that i² = j² = k² = ijk = -1. A quaternion then is a number of the form a + bi + cj + dk, where a, b, c, and d are real numbers uniquely determined by the quaternion. The multiplication of quaternions could be deduced from the following multiplication table:

	1	i	j	k
1	1	i	j	k
i	i	-1	k	-j
j	j	-k	-1	i
k	k	j	-i	-1

These products form the quaternion group of order 8, Q₈.

Unlike real or complex numbers, multiplication of quaternions is not commutative: ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. The quaternions are an example of a skew field, 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. They 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 skew fields over the field of real numbers.

The conjugate of the quaternion z = a + bi + cj + dk is defined as z^* = a - bi - cj - dk, and the absolute value of z is defined by |z|² = zz^* = a² + b² + c² + d². Note that (wz)^*=z^*w^* but not equals w^*z^* in general. By using the distance function d(z,w) = |z - w|, the quaternions form a metric space 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.

The set of quaternions of absolute value 1 forms a 3-dimensional sphere S³ and a group (even a Lie group) under multiplication. This group acts by conjugation on the copy of R³ consisting of quaternions with real part equal to zero: it is not hard to see that the conjugation by a unit quaternion of real part cos t is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part. Thus, S³ is the double cover of the group SO(3) of real orthogonal 3x3 matrices of determinant 1; it is isomorphic to SU(2), the group of complex unitary 2x2 matrices of determinant 1.

Let A be the set of quaternions of the form a + bi + cj + dk where a, b, c and d are either all integers or all rational numbers with odd numerator and denominator 2. The set A is a ring and a lattice. There are 24 unit quaternions in this ring and they are the vertices of a regular polytope called {3,4,3} in Schlafli's notation.

Quaternions are sometimes used in computer graphics (and associated geometric analysis) to represent the orientation of an object in 3d space. The advantages are: non singular representation (compared with Euler angles for example), more compact (and faster) than matrices. (See quaternions and spatial rotation.)

Representing quaternions by matrices

There are at least two ways of representing quaternions as matrices. One is to use 2x2 complex matrices, and the other is to use 4x4 real matrices.

In the first way, the quaternion a + bi + cj + dk is represented as:

{\begin{pmatrix}a-di&-b+ci\\b+ci&a+di\end{pmatrix}}

A nice property of this representation is that all complex numbers are matrices with only real entries.

In the second way, the quaternion a + bi + cj + dk is represented as:

{\begin{pmatrix}a&-b&d&-c\\b&a&-c&-d\\-d&c&a&-b\\c&d&b&a\end{pmatrix}}

History

Quaternions were discovered by William Rowan Hamilton 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 a story he told, he was out walking one day with his wife when the solution in the form of equation i² = j² = k² = ijk = -1 suddenly occurred to him; he then promptly carved this equation into the side of nearby Brougham bridge.

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.

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 i² = a, j² = b and ij = -ji. These algebras are either isomorphic to the algebra of 2-by-2 matrices over F, or they are division algebras over F. They are called quaternion algebras.