A Glimpse of Quaternion and A Few Discussions

Quaternion is a very interesting mathematical concept that I had never touched before I reviewed the source code in the recent project I've been involved in.

It is a number system that extends complex numbers discovered by the great mathematician Sir William Rowan Hamilton in 1843 and is no less impressive than any other wellknown discoveries he made.

It was used as theoretical and engineering tools intensively and even compulsorily during the days after its discovery especially in Ireland and Britain where the mathematician oringinated. And it was for a long time replaced by more easy-to-use tools like matrix and vector analysis and so forth, until late 20 century with the advent of new areas of mathematics and technologies, it started drawing attention from people again. As particularly for computer graphics, it may provide a more efficient way to implement object rotation than the normal matrix based way does. And as it seems to me its significance in mathematics world may also largely lie in its position in the algebra theory, as it is 'one of the only two finite-dimensional division rings containing real numbers as a proper string'. I am quite interested in that but we can only see why it is after we have a clear idea of what quaternion is.

In saying its extending complex number system it means that it can be seen as a number systems that can be represented geometrically in high-dimensional euclid spaces as complex numbers are in a 2-dimensional one.

== Multiplication of Quaternion ==

As a set, quaternion is equal to R^4. A quaternion numbr as an element in the set is normally represented as (a + b i + c j + d k) in which a, b, c and d are real numbers and  1, i, j and k serves as a set of bases in space R^4 as the quaternion system is equal to. In this sense, the set has all linear space properties. And a multiplication operation is additionally defined on it which makes quaternion different from conventional R^4 space. The defintion of the operation is started by the definintion of the basis historically and cognitively:

i^2 = j^2 = k^2 = ijk = -1; multiplication with all real numbers (including -1) is commutible.

Based on the muplication defined on the system which is non-commutative but is associative and distributes over addition, the following can easily be deduced:

jk = i
ij = k

the rest of the multiplication of basis are a little bit less straightforward,

first by right multiplying k by i, it gives

ki = ijjk = -ik

right multiply it by ik, we have:

1= kiik = -ikik

left multiplied by i, we have：

i = kik

left multiplied by j, we have:

ji = jkik = iik = -k, therefore ji = -k

right multiplied by j, we have

i = jk = jij = -kj, therefore kj = -i

similarly, ki = j, ik = -j

When muliplying two arbitrary quaternion numbers, and if the quaternion is represented in the tuple of a real number and a 3D vector as (r, v), it gives

(r1, v1) (r2, v2) = r1 r2 - v1·v2 + r1 v2 + r2 v1 + v1×v2

the result of the muplication gives inner and outer products of two vectors in a uncluttered manner, and one can easily retrieve these two results by setting r1 and r2 to zero.

== Division of Quaternion ==

If a quaternion is expressed as q = a + b i + c j + d k, its conjugate is defined as q* = a - b i - c j - d k.

Unlike complex numbers, conjugate of quaternion can be expressed entirely with multiplication and addition,

q* = -0.5 (q + iqi + jqj + kqk)

q q* = q* q = || q ||^2

The reciprocal can be defined as q^-1 = q* / || q ||^2, the multiplication of a quaternion and its reciprocal in either order is 1.

Like multiplication, there are two types of division, the quotien can either be:

p^-1 q and q p^-1 which are not necessariliy equal.

== Algebraic Properties ==

With the operations defined above, we can now have a look at the algebraic properties of quaternion:

Since quaternion is divisible, it forms a division algebra[2] ，which is similar to field （域） structure with division enabled but without commutativity of multiplication. Or it can be said as a non-commutative associative algebra over real numbers. Frobenious theorem [5] states that real numbers R, complex numbers C, and Quaternion Q are the only three finite-dimensional division algebra over real numbers.

Quaternion is also made into a normed algebra [6], which over the reals are also very rare, it's also an example of a composition algebra and of a unital Banach algebra. (this will be discussed in detail later when these concepts have been clarified)

Group based on set {+/-1, +/-i, +/-j, +/-k} under multiplication (non-commutative) is called Quaternion group denoted Q8. The group ring [8] of Q8 is ring RQ8 which is also an 8-dimensional vector space over R, with elements in the form of a + b(-1) + c i + d (-i) + e j + f (-j) + g k + h (-k).
The quaternions are the quotient ring of RQ8 by the ideal generated by the elements 1+(-1), i+(-i), j+(-j), k+(-k). (To be proved)

== Appendix A A bit revision of algebra ==

A more detailed discussion on topics of abstract algebra will be performed in separate articles since abstract algebra by far is a subject of high intricacy and abstrusness.

1. Set, a collection of distinct elements with no operation, function or relationship defined on them.

2. Magma （代数）, a set that a binary operation is defined on its elements and the result of the operation is always an element of the set.

3. Semigroup （半群）, if the operation (denoted by ·) defined in Magma is associative, i.e., for all a, b and c within the set it always holds that (a · b) · c = a · (b · c)

4. Monoid （单群）, if unique identity element exists in the semigroup, i.e. there is an element denoted by 1, that for all x within the semigroup it always holds that 1 · x = x · 1 = x

5. Quasigroup（类群）, if the magma is divisible, i.e. for each a and b in the magma, there exists unique elements x and y in the magma such that
    a · x = b = y · a. So two types of division operations can be defined as x = a \ b and y = b / a respectively being left and right division.

6. Loop（圈，不是环！）, if a quasigroup has unique identity element, it is a loop.

7. Group（群）, if a manoid has invertibility or equivalently a loop has associativity, it is a group

8. Abelian Group（阿贝尔群）, if a group is commutive, it's a Abelian group

9. Pseudo Ring（伪环）, a set forms abelian group under operation + and semigroup under ·, with distributivity of · over +. Note pseudo ring is referred to as ring in book [7], however this disparity doens't usually cause big problems.

10. Ring（环）, Pseudo Ring with the semigroup under · being monoid (having identity element 1), e.g. Square matrices all of the same dimensions.

11. Commutative Ring（交换环）, Ring with commutativity of ·.

12. Integral Domain（整环）, Ring with commutivity and identity of · and having no zero divisor for ·. which means for all non-zero element a in the set, there is no such non-zero element b that a · b = 0, or b · a = 0. All integers form an integral domain, while integral matrices don't and '{ x_ = x mod M | x in Z } with x_ + y_ = (x+y) mod M and x_ · y_ = (x·y) mod M ' when M is not a prime number is not an integral domain （because in this case M is the product of two values less than it namely a and b, then a_ · b_ = 0_)

13. Field（域）, Commutitative ring with inverses for ·. (e.g. rational numbers, real numbers, complex numbers)

A varity of computer software supports algebra operation and demonstration, including the well-known mathematica and a system developed by University of Sydney dedicated to abstract algebra computation given in the reference list [4].

14. Group ring （群生成环?）A ring formed by the mappings from group G to ring R denoted R[G] defined as: f: G->R of finite support, where the product af of a scalar a in R and a mapping f is defined as a mapping: x -> a · f(x), and the sum of two mappings f and g is x->f(x) + g(x). To turn the commutative group into a ring, the product of its elements is such a mapping that: x -> sigma{all uv=x, u,v in G; f(u) · g(v)}.

References:点击打开链接

[1] http://en.wikipedia.org/wiki/Quaternion

[2] http://en.wikipedia.org/wiki/Division_algebra

[3] Introductory lessons on algebra for beginners on Youtube http://www.youtube.com/user/VeritySeeker

[4] Magma (abstract algebra simulating and computational software system) developed by

[5] Frobenious theorem http://en.wikipedia.org/wiki/Frobenius_theorem_%28real_division_algebras%29

[6] Banach Algebra http://en.wikipedia.org/wiki/Normed_algebra

[7] 应用近世代数 胡冠章著

[8] Group ring http://en.wikipedia.org/wiki/Group_ring