Quaternion

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	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:

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. 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:

<math>\begin{pmatrix} a-di & -b+ci \\ b+ci & a+di \end{pmatrix}</math>

This representation has several nice properties:

All complex numbers (c = d = 0) correspond to matrices

with only real entries.

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 S³ and SU(2). The latter group is important in quantum mechanics when dealing with spin; see all Pauli matrices.

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

<math>\begin{pmatrix} a & -b & d & -c \\

 b & a & -c & -d \\
 -d & c & a & -b \\
 c & d & b & a \end{pmatrix}</math>

In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix.

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.

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 four-element multiple of real numbers, and described the first element 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.

Even by this time there was controversy about the use of quaternions. Some of Hamilton's supporters viciously opposed the growing fields of vector algebra and vector calculus (from developers like Oliver Heaviside and Willard Gibbs), maintaining that quaternions provided a superior notation. While this may be true in three dimensions plus time (i.e., spacetime), quaternions cannot be used in other dimensions (though other deriverative exist like Octonions and Clifford algebras for this). Their scientific recognition compared to vectors has therefore decreased over time. They are today still used in computer graphics and Plasma physics.

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[?].

Related resources

Doing Physics with Quaternions (http://world.std.com/~sweetser/quaternions/qindex/qindex.html)
The Physical Heritage of Sir W. R. Hamilton (http://arxiv.org/pdf/math-ph/0201058) (PDF)

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