1

Quaternion Algebra

This chapter introduces the quaternion algebra and presents some properties that will be useful in later chapters.

1.1. Definitions

Quaternions are one of the four existing normed division algebras over the real numbers. Classically denoted by in honor of Sir W.R. Hamilton who discovered them in 1843, they form a non-commutative algebra. A quaternion q ∈ is a four-dimensional (4D) hypercomplex number and has a Cartesian form given by:

[1.1]

where a, b, c, d ∈ are called its components. The three imaginary units i, j, k are square roots of –1 and are related through the famous1 relations:

[1.2]

A quaternion q ∈ can be decomposed into a scalar part S(q) and a vector part V(q):

[1.3]

where S(q) = a and V(q) = q – S(q) = ib + jc + kd. Obviously, S(q) ∈ and we will also refer to it as the real part of q, i.e. (q) = a. Now, q ∈ will be called a pure quaternion if its real part is null, i.e. if S(q) = 0. The set of pure quaternions will be denoted as V(), while the set of quaternions with a null vector part are trivially identified with elements of , i.e. S() ≡ . If q has a null vector part, V(q) = 0, then q is simply an element of . To identify the different imaginary components of a quaternion q = a + ib + jc + kd, we will use the following notations:

[1.4]

so that a quaternion q ∈ can be written as:

[1.5]

The Cartesian notation for a quaternion q ∈ is, in fact, its expression in a specific 4D basis of the algebra , namely in the basis {1, i, j, k}. Recall that, as an algebra, possesses a vector space structure that allows the expression of any of its elements in terms of its components in a basis of . The basis {1, i, j, k} is the most common and popular basis to express a quaternion. However, we may encounter some other bases later on in this book, leading to alternate notations for a quaternion q ∈ . Before introducing these notations, we first review some remarkable properties of quaternions.

1.2. Properties

Here, we list some of the properties of quaternions that will be used throughout the book.

From the algebra structure of , the sum of two quaternions is trivial. Given two quaternions q and p, we have:

[1.6]

Expressing the two quaternions in their Cartesian forms, q = a + ib + jc + kd and p = e + if + jg + kh, their sum is:

[1.7]

and their product takes the form:

[1.8]

Using the scalar/vector notation, this product takes the following form:

[1.9]

where 〈.,.〉 is the scalar product and × is the vector cross product. These are understood in the classical sense of the three-dimensional (3D) vector cross and inner products, which means that:

[1.10]

which is scalar valued, i.e. 〈V(q), V(p)〉 ∈ , and that:

[1.11]

where the result is a pure quaternion, i.e. (V(q) × V(p)) ∈ V().

A very noticeable property is that the product of two quaternions is not commutative so that in general:

[1.12]

This can be inferred from the presence of the non-commutative cross product in [1.9]. Note, however, that the product of quaternions is associative so that for any three quaternions q, p, r ∈ , the following is true:

[1.13]

The norm of a quaternion q is defined as:

[1.14]

A quaternion q ∈ with ||q|| = 1 is said to be a unit quaternion. As previously mentioned, is one of the four existing normed division algebras. As a result, given any two quaternions p, q ∈ , then:

[1.15]

It can also be easily checked that ||qp|| = ||pq||. A related quantity that will be used in the following is the modulus of a quaternion. It is defined as the length of the quaternion in Euclidean 4D space. The modulus of q ∈ is denoted by |q| and is expressed as:

[1.16]

Obviously, |q| ∈ + and |q| = 0 if and only if q = 0. Like the norm, the modulus of a product of two quaternions p and q has the following property:

[1.17]

Just as with the complex numbers, the conjugate of a quaternion q is obtained by negating its imaginary part. However, in the imaginary part is 3D and consists of the entire vector part V(q). Denoted by , the conjugate of q is thus defined as:

[1.18]

It follows that the scalar and vector parts of any quaternion q ∈ can be obtained by:

[1.19]

Conjugation in has the following property:

[1.20]

In contrast to the complex case, conjugation is not an involution2 but an antiinvolution, such that for q, p ∈ :

[1.21]

that is, the order of the factors in a quaternion product is reversed by the conjugation operator. Note that the modulus (and also the norm) of a quaternion q can be expressed using the conjugate of q as:

[1.22]

It is also possible to define involutions over . Involutions are defined with respect to a pure unit quaternion μ (μ ∈ V() and |μ| = 1). The most general case (together with many properties) is presented in [ELL 07d]. As special cases, one can choose μ as one of the standard basis elements of , i.e. i, j or k. Involutions with respect to these three unit pure quaternions can be called canonical. Given a quaternion q, its three canonical involutions are:

[1.23]

Clearly, q and its three canonical involutions allow us to recover the four components a, b, c, d of q by linear combination. Now, the most general definition for involution is:

[1.24]

where μ ∈ V() and |μ| = 1.

Involutions in possess many properties (see [ELL 07d] for details) among which, for q, p ∈ and μ ∈ V() and μ2 = –1 (i, j and k are possible choices for μ):

[1.25]

As is a division algebra, any non-null quaternion possesses an inverse. The inverse of a given quaternion q ∈ is given by:

[1.26]

where it can be easily checked that qq–1 = 1 because of [1.22]. Note that for a pure unit quaternion μ, (||μ|| = 1 and S(μ) = 0), the following holds: μ–1 = –μ.

Now that we have introduced the inverse of a quaternion, we are ready to look at the ratio of two quaternions p and q. Ratios must be handled with care in (indeed, in any non-commutative algebra), and it is preferable to avoid the p/q notation when possible, as it is ambiguous, since p/q can be interpreted as the product of p by q–1; the notation p/q does not specify the order of the product so that it leaves the possibility for pq–1 and q–1p. The ambiguity arises from the fact that in general:

[1.27]

The above non-equality arises from the fact that:

[1.28]

Thus, it is important to consider ratios as products with the inverse and to take care of the order of the product.

Now, the modulus of a ratio is an interesting quantity to consider, as it does not suffer from the order in the multiplication, due to the property of the modulus given in [1.17]. It thus follows that:

[1.29]

It can be useful to write a given quaternion q ∈ as a product of a scalar positive number (its modulus) and a unit quaternion. This can be done in the following way:

[1.30]

where we used the notation for the unit modulus version of q, i.e. = q/ |q| so that || = 1. Note that the decomposition of q into the product of its modulus and its normalized version is unique. The normalized version of q, denoted by , is also sometimes called a versor. Finally, it must be emphasized that if q is a pure quaternion, i.e. q ∈ V(), then it is uniquely written as q = |q| μ where we have denoted = μ to highlight the fact that it is a pure unit quaternion.

In [1.5], we introduced the Cartesian form of a quaternion q ∈ , in which it is expressed using the sum of a real part (q), an i–imaginary part i(q), a j–imaginary part j(q) and a k–imaginary part k(q). This expression is a special case of the expansion of a quaternion over a 4D basis. The specific basis used in [1.5] is {1, i, j, k}. This is the classical basis used by most authors. Now, it is possible to use a different basis and it turns out that there is an infinite amount of choices for a basis in . Given two pure unit quaternions...