QUATERNIONS

In mathematics, the quaternions are a number system that extends the complex numbers
. They were first described by Irish mathematician William Rowan Hamilton in 1843^[1][2] and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space^[3] or equivalently as the quotient of two vectors.^[4]

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and computer vision. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them depending on the application.

In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered.^[5] The algebra of quaternions is often denoted by H (for Hamilton), or inblackboard bold by  (Unicode U+210D, ℍ). It can also be given by the Clifford algebra classifications Cℓ_0,2(R) ≅ Cℓ⁰_3,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra.

The unit quaternions can therefore be thought of as a choice of a group structure on the 3-sphere S³ that gives the group Spin(3), which is isomorphic to SU(2) and also to theuniversal cover of SO(3).

Definition[

As a set, the quaternions H are equal to R⁴, a four-dimensional vector space over the real numbers. H has three operations: addition, scalar multiplication, and quaternion multiplication. The sum of two elements of H is defined to be their sum as elements of R⁴. Similarly the product of an element of H by a real number is defined to be the same as the product by a scalar in R⁴. To define the product of two elements in H requires a choice of basis for R⁴. The elements of this basis are customarily denoted as 1, i, j, and k. Every element of H can be uniquely written as a linear combination of these basis elements, that is, as a1 + bi + cj + dk, where a, b, c, and d are real numbers. The basis element 1 will be the identity element of H, meaning that multiplication by 1 does nothing, and for this reason, elements of H are usually written a + bi + cj + dk, suppressing the basis element 1. Given this basis, associative quaternion multiplication is defined by first defining the products of basis elements and then defining all other products using the distributive law.

Multiplication of basis elements

The identities

,

where i, j, and k are basis elements of H, determine all the possible products of i, j, and k.

For example right-multiplying both sides of −1 = ijk by k gives

All the other possible products can be determined by similar methods, resulting in

which can be expressed as a table whose rows represent the left factor of the product and whose columns represent the right factor, as shown at the top of this article.

Noncommutativity of multiplication

Noncommutativity of quaternion multiplication

×	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

Unlike multiplication of real or complex numbers, multiplication of quaternions is not commutative. For example, ij = k, while ji = −k. The noncommutativity 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 z² + 1 = 0, for instance, has infinitely many quaternion solutions z = bi + cj + dk with b² + c² + d² = 1, so that these solutions lie on the two-dimensional surface of a sphere centered on zero in the three-dimensional subspace of quaternions with zero real part. This sphere intersects the complex plane at two points i and −i.

The fact that quaternion multiplication is not commutative makes the quaternions an often-cited example of a strictly skew field.

Hamilton product

For two elements a₁ + b₁i + c₁j + d₁k and a₂ + b₂i + c₂j + d₂k, their product, called the Hamilton product (a₁ + b₁i + c₁j + d₁k) (a₂ + b₂i + c₂j + d₂k), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

Now the basis elements can be multiplied using the rules given above to get:^[6]

Ordered list form

Using the basis 1, i, j, k of H makes it possible to write H as a set of quadruples:

Then the basis elements are:

and the formulas for addition and multiplication are:

Scalar and vector parts

A number of the form a + 0i + 0j + 0k, where a is a real number, is called real, and a number of the form 0 + bi + cj + dk, where b, c, and d are real numbers, and at least one of b,c or d is nonzero, is called pure imaginary. If a + bi + cj + dk is any quaternion, then a is called its scalar part and bi + cj + dk is called its vector part. The scalar part of a quaternion is always real, and the vector part is always pure imaginary. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to define a vector to mean a pure imaginary quaternion. With this convention, a vector is the same as an element of the vector space R³.

It is important to note, however, that the vector part of a quaternion is, in truth, an "axial" vector or "pseudovector", not an ordinary or "polar" vector, as was formally proven by S.L. Altmann in Ch. 12 of his 1986 book, "Rotations, Quaternions and Double Groups". A polar vector can be represented in calculations (for example, when rotated by a quaternion "similarity transform") by a pure quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180 deg) rotation quaternion corresponds to the direction of the represented polar vector in such a case.

Hamilton called pure imaginary quaternions right quaternions^[15][16] and real numbers (considered as quaternions with zero vector part) scalar quaternions.

If a quaternion is divided up into a scalar part and a vector part, i.e.

then the formulas for addition and multiplication are:

where "·" is the dot product and "×" is the cross product.

Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let  be a quaternion. The conjugate of q is the quaternion . It is denoted by q^∗, q,^[6] q^t, or . Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq)^∗ = q^∗p^∗, not p^∗q^∗.

Unlike the situation in the complex plane, the conjugation of a quaternion can be expressed entirely with multiplication and addition:

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of p is (p + p^∗) / 2, and the vector part of p is (p − p^∗) / 2.

The square root of the product of a quaternion with its conjugate is called its norm and is denoted ||q|| (Hamilton called this quantity the tensor of q, but this conflicts with modern meaning of "tensor"). In formula, this is expressed as follows:

This is always a non-negative real number, and it is the same as the Euclidean norm on H considered as the vector space R⁴. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then

This is a special case of the fact that the norm is multiplicative, meaning that

for any two quaternions p and q. Multiplicativity is a consequence of the formula for the conjugate of a product. Alternatively it follows from the identity

(where i denotes the usual imaginary unit) and hence from the multiplicative property of determinants of square matrices.

This norm makes it possible to define the distance d(p, q) between p and q as the norm of their difference:

This makes H into a metric space. Addition and multiplication are continuous in the metric topology.

Unit quaternion

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion Uq called the versor of q:

Every quaternion has a polar decomposition .

Using conjugation and the norm makes it possible to define the reciprocal of a quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of q and q^∗/ (in either order) is 1. So the reciprocal of q is defined to be

This makes it possible to divide two quaternions p and q in two different ways. That is, their quotient can be either p q⁻¹ or q⁻¹p. The notation p/q is ambiguous because it does not specify whether q divides on the left or the right.

Algebraic properties

Cayley graph of Q₈. The red arrows represent multiplication on the right byi, and the green arrows represent multiplication on the right by j.

The set H of all quaternions is a vector space over the real numbers with dimension 4. (In comparison, the real numbers have dimension 1, the complex numbers have dimension 2, and the octonions have dimension 8.) Multiplication of quaternions, for example, is associative and distributes over vector addition, but it is not commutative. Therefore, the quaternions H are a non-commutative associative algebraover the real numbers. Even though H contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The Frobenius theorem states that there are exactly three: R, C, and H. The norm makes the quaternions into a normed algebra, and normed division algebras over the reals are also very rare: Hurwitz's theorem says that there are only four: R, C, H, and O (the octonions). The quaternions are also an example of a composition algebra and of a unital Banach algebra.

Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±j, ±k} forms a group under multiplication. This group is called the quaternion group and is denoted Q₈.^[17] The real group ring of Q₈ is a ring R[Q₈] which is also an eight-dimensional vector space over R. It has one basis vector for each element of Q₈. The quaternions are the quotient ring of R[Q₈] by the ideal generated by the elements 1 + (−1), i + (−i), j + (−j), and k + (−k). Here the first term in each of the differences is one of the basis elements 1, i, j, and k, and the second term is one of basis elements −1, −i, −j, and −k, not the additive inverses of 1, i, j, and k.