Quaternion

Definition

While the complex numbers are obtained by adding the element i to the real numbers which satisfies , the quaternions are obtained by adding the elements i, j and k to the real numbers which satisfy the following relations.

Every quaternion is a real linear combination of the basis quaternions 1, i, j, and k, i.e. every quaternion is uniquely expressible in the form . In other words, as a vector space over the real numbers, the set H of all quaternions has dimension 4, whereas the complex number plane has dimension 2. 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 basis quaternions; this table is given at the right. Under this multiplication, the basis quaternions, with their negatives, form the quaternion group of order 8, Q₈.The scalar part of the quaternion is a while the remainder is the vector part.Thus a vector in the context of quaternions has zero for scalar part.

Example

Let

x = 3 + i
:y = 5i + j - 2k

x + y = 3 + 6i + j - 2k
:xy = (3 + i)(5i + j - 2k)
::= 15i + 3j - 6k + 5i² + ij - 2ik
::= 15i + 3j - 6k - 5 + k + 2j
::= - 5 + 15i + 5j - 5k
:yx = (5i + j - 2k)(3 + i)
::= 15i + 5i² + 3j + ji - 6k - 2ki
::= 15i - 5 + 3j - k - 6k - 2j
::= - 5 + 15i + j - 7k

Properties

:
:
:
:
:

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 , for instance, has the infinitely-many quaternion solutions with . The conjugate of the quaternion 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 R⁴) 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.

Fundamental formula

Profile

From this set equality one can view H as the union of complex planes sharing the same real line and taking an imaginary unit from the set.Furthermore, the unit sphere in H, the 3-sphere, is formed by the collection of unit circles in these complex planes.As the general point on a circle is

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 R³ 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:

1. Non singular representation (compared with Euler angles for example)
2. More compact (and faster) than matrices
3. 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 S³ and SU(2). The latter group is important in quantum mechanics when dealing with spin; see also Pauli matrices.

Quaternion operations

Quaternion operations have extended applications in electrodynamics and general relativity. The use of quaternions can replace tensors in representation. It is sometimes easier to use quaternions with complex elements, leading to a form that is not a division algebra. However, the same operations can be performed using a combination of conjugate operations. Only quaternions with real elements will be discussed here. The discussion will involve describing quaternions in two forms. One as a combination of a vector and a scalar, and the other as a combination of the two constructors and the bivector (i, j, and k).

Define two quaternions:
:
:

where represents the vector , and represents the vector .

Addition, Products, and General functions

Quaternion addition -
p + q :
Like complex numbers, vectors, and matrices, the addition of two quaternions is equivalent to summing the elements together:

Addition follows all of the commutativity and associativity rules of real and complex number.

Quaternion multiplication -
pq :
The usual non-commutative multiplication between two quaternions is termed the Grassmann product. This product has been described briefly above. The complete form is described below:

Due to the non-commutative nature of the quaternion multiplication, pq is not equivalent to qp. The Grassmann product is useful to describe many other algebraic functions. The vector portion of the multiplication of qp follows:

Quaternion dot-product -
p · q :

The dot-product is also referred to as the Euclidean inner product, and is equivalent to a 4-vector dot product. The dot product is the sum of the quantity of each element of p multiplied by each element of q. It is a commutative product between quaternions, and returns a scalar quantity.

The dot-product can be rewritten using the Grassmann product:

This product is useful to find isolate an element from a quaternion. For instance, the i term can be pulled out from p:

Quaternion outer-product -
Outer(p,q) :
The Euclidean outer-product is not used often; however, it is mentioned as a pair with the inner-product because of the similarity in the Grassmann product form:

Quaternion even-product -
Even(p,q) :
The even-product of quaternions is also not widely used, but it mentioned due to the similarity between it and the odd product. It is the purely symmetric product; therefore, it is completely commutative.

Quaternion cross-product -
p × q
The cross-product of quaternions is also known as the odd-product. It is equivalent to the vector cross-product, and returns a vector quantity only:

Quaternion inverse -
p⁻¹
The inverse of a quaternion is defined in a way that p⁻¹p = 1. It is defined above in the definition section, under properties (note the difference in variable notation). It is formed the same way that the complex inverse is found:

The dot product of a quaternion is a scalar. The division of a quaternion by a scalar is equivalent to multiplication by the scalar inverse, such that each element of the quaternion is divided by the divisor.

Quaternion division -
p⁻¹q
The non-commutativity of quaternions allows for two divisions of numbers p⁻¹q and qp⁻¹. This means that the notation of q/p cannot be used unless p is a scalar only.

Quaternion scalar -
Scalar(p) :
The scalar of a quaternion can be isolated in the same way that was described earlier with the dot-product:

Quaternion vector -
Vector(p) :
The vector of a quaternion can be isolated using the outer-product in the same way the inner product is used to isolate the scalar:

Quaternion modulus -
:
The absolute value of a quaternion is the scalar quantity that determines the length of the quaternion from the origin.

Quaternion sign -
sgn(p) :
The sign of a complex number finds the complex number of the same direction found on the unit circle. The quaternion sign also produces the unit quaternion:

Quaternion argument -
arg(p) :
The argument finds the angle of the 4-vector quaternion from the unit scalar (i.e. 1). This returns a scalar angle.

Exponentials and Logarithms

• Natural exponential:

• Natural logarithm:

• Power:
  

  Trigonometry

• Sine:

• Cosine:

• Tangent:
  

  Hyperbolic

• Hyperbolic sine:

• Hyperbolic cosine:

• Hyperbolic tangent:
  

  Inverse Hyperbolic functions

• Inverse hyperbolic sine:

• Inverse hyperbolic cosine:

• Inverse hyperbolic tangent:
  

  Inverse trigonometric functions

  
  This was listed last, because the inverse hyperbolic functions needed to be defined over quaternions first.

• Inverse sine:

• Inverse cosine:

• Inverse tangent:
  

  Construction of quaternions from complex numbers

  
  According to the Cayley-Dickson construction, a quaternion is an ordered pair of complex numbers. Letting j be a new root of −1, different from both i and −i, and given u and v are a pair of complex numbers, then
  :
  is a quaternion.

With these rules, we can now derive the multiplication table for i, j and i j, the imaginary components of a quaternion:
:
:
:
:
:
:
:
:
:
Notice how the dyad i j behaves just like the k in the definition.

For any complex number v = c + i d, its product with j has the following property:
:
since
:.

Let p be the quaternion with complex components w and z:
:.
Then the product q p is
:
::
::
Since the product of complex numbers is commutative, we have
:
which is precisely how quaternion multiplication is defined by the Cayley-Dickson construction.

Note that if u = a + i b, v = c + i d, and p = a + i b + j c + k d then p′s construction from u and v is rather
:.

Generalizations

History

Quaternions were introduced by Sir 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, he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

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 directly applied in higher dimensions (though extensions like octonions and Clifford algebras may be more applicable). Vector notation had nearly universally replaced quaternions in science and engineering by the mid-20th century.

Some early formulations of Maxwell's equations used a quaternion-based notation (although Maxwell's original formulation simply used 20 equations in 20 variables), but it proved unpopular compared to the vector-based notation of Heaviside. (All of these formulations were mathematically equivalent.)

Recent years

Quaternions are often used in computer graphics (and associated geometric analysis) to represent rotations (see quaternions and spatial rotation) and orientations of objects in 3d space. They are smaller than other representations such as matrices, and operations on them such as composition can be computed more efficiently. 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, avoiding such phenomena as gimbal lock.

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 Hamilton's carving remains.

See also