## Quaternion

A member of a noncommutative Division Algebra first invented by William Rowan Hamilton. The quaternions are sometimes also known as Hypercomplex Numbers and denoted . While the quaternions are not commutative, they are associative.

The quaternions can be represented using complex Matrices

 (1)

where and are Complex Numbers, , , , and are Real, and is the Complex Conjugate of . By analogy with the Complex Numbers being representable as a sum of Real and Imaginary Parts, , a quaternion can also be written as a linear combination
 (2)

of the four matrices
 (3) (4) (5) (6)

(Note that here, is used to denote the Identity Matrix, not .) The matrices are closely related to the Pauli Spin Matrices , , , combined with the Identity Matrix. From the above definitions, it follows that
 (7) (8) (9)

Therefore , , and are three essentially different solutions of the matrix equation
 (10)

which could be considered the square roots of the negative identity matrix.

In , the basis of the quaternions can be given by

 (11) (12) (13) (14)

The quaternions satisfy the following identities, sometimes known as Hamilton's Rules,

 (15)

 (16)

 (17)

 (18)

They have the following multiplication table.

 1 1 1

The quaternions , , , and form a non-Abelian Group of order eight (with multiplication as the group operation) known as .

The quaternions can be written in the form

 (19)

The conjugate quaternion is given by
 (20)

The sum of two quaternions is then
 (21)

and the product of two quaternions is
 (22)

so the norm is
 (23)

In this notation, the quaternions are closely related to Four-Vectors.

Quaternions can be interpreted as a Scalar plus a Vector by writing

 (24)

where . In this notation, quaternion multiplication has the particularly simple form

 (25)

Division is uniquely defined (except by zero), so quaternions form a Division Algebra. The inverse of a quaternion is given by
 (26)

and the norm is multiplicative
 (27)

In fact, the product of two quaternion norms immediately gives the Euler Four-Square Identity.

A rotation about the Unit Vector by an angle can be computed using the quaternion

 (28)

(Arvo 1994, Hearn and Baker 1996). The components of this quaternion are called Euler Parameters. After rotation, a point is then given by
 (29)

since . A concatenation of two rotations, first and then , can be computed using the identity
 (30)

(Goldstein 1980).

See also Biquaternion, Cayley-Klein Parameters, Complex Number, Division Algebra, Euler Parameters, Four-Vector, Octonion

References

Altmann, S. L. Rotations, Quaternions, and Double Groups. Oxford, England: Clarendon Press, 1986.

Arvo, J. Graphics Gems II. New York: Academic Press, pp. 351-354 and 377-380, 1994.

Baker, A. L. Quaternions as the Result of Algebraic Operations. New York: Van Nostrand, 1911.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 107, Feb. 1972.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 230-234, 1996.

Crowe, M. J. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. New York: Dover, 1994.

Dickson, L. E. Algebras and Their Arithmetics. New York: Dover, 1960.

Du Val, P. Homographies, Quaternions, and Rotations. Oxford, England: Oxford University Press, 1964.

Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R. Numbers. New York: Springer-Verlag, 1990.

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 151, 1980.

Hamilton, W. R. Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method. Dublin: Hodges and Smith, 1853.

Hamilton, W. R. Elements of Quaternions. London: Longmans, Green, 1866.

Hamilton, W. R. The Mathematical Papers of Sir William Rowan Hamilton. Cambridge, England: Cambridge University Press, 1967.

Hardy, A. S. Elements of Quaternions. Boston, MA: Ginn, Heath, & Co., 1881.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Cambridge, England: Clarendon Press, 1965.

Hearn, D. and Baker, M. P. Computer Graphics: C Version, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 419-420 and 617-618, 1996.

Joly, C. J. A Manual of Quaternions. London: Macmillan, 1905.

Julstrom, B. A. Using Real Quaternions to Represent Rotations in Three Dimensions.'' UMAP Modules in Undergraduate Mathematics and Its Applications, Module 652. Lexington, MA: COMAP, Inc., 1992.

Kelland, P. and Tait, P. G. Introduction to Quaternions, 3rd ed. London: Macmillan, 1904.

Nicholson, W. K. Introduction to Abstract Algebra. Boston, MA: PWS-Kent, 1993.

Tait, P. G. An Elementary Treatise on Quaternions, 3rd ed., enl. Cambridge, England: Cambridge University Press, 1890.

Tait, P. G. Quaternions.'' Encyclopædia Britannica, 9th ed. ca. 1886. ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/tait/Encyc-Brit.ps.gz.