User:Robotics lab/context

Contexts of quaternion geometry and linear algebra

A source of great confusion is that quaternion geometry and multi linear algebra are two completely different contexts. Multi linear algebra and classical quaternion nomenclature share many of the same words, and both can be used to study the geometry of space time. But the same word can often mean two very different things in the two different contexts. This section explains the various points of view related to these different contexts.

Three important examples of words that have both a linear algebra context and a classical quaternion geometry context are vector, tensor and space . Algebra and even the word quaternion have taken on different meanings in these two different branches of science.

Linear algebra exclusive point of view

Contrary to the classical quaternion point of is the view that some extension of linear algebra is best suited for all problems both systems attempt to address. This point of view seeks to extend algebra by extracting ideas from the old classical quaternion system and then proclaim that classical quaternion geometry is musty, obsolete and unnecessary, and has been replaced by some new extension of algebra. This view existed in both the 19th and 20th century, and still exists in the 21st.

A factor related to this view that makes it difficult for some beginners to understand commentary written from a linear algebra exclusive point of view on classical quaternion thinking is the use of Quaternion negative nomenclatures. Modern quaternion negative nomenclatures tend to use words with negative connotations for classical quaternion concepts. For an example of the use of quaternion negative terms consider the phrase 'a type of complex number' in reference to a versor.

On the other hand quaternion negative nomenclatures tend to use positive terms for concepts developed in the context of linear algebra. One of the most glaring examples of the built in bias of these quaternion negative nomenclatures is calling a entity consisting exclusively of three-tuples of numbers with a time like quality who have a square of plus one real, and numbers with a distance like quality that have a square of minus one such as classical vectors imaginary.

Classic isomorphism tests

Isomorphic means of the same shape. Extensions to linear algebra often claim to have discovered a new system that is 'isomorphic' to the quaternions. These claims can be confusing because they often involve first redefining the classical quaternion and then introducing the new entity that is actually isomorphic to this redefined quaternion.

The Tait test

The Tait test is to count the number of kinds of multiplication in a system claiming to be isomorphic to the quaternions. If there is more than one cardinal type of multiplication from the point of view of the important classical thinkers of the later period lead by Peter Guthrie Tait the entity is not a classical quaternion. 19th century debate raged over this issue with regard to early versions of the notion of a normed vector space over the reals lead by Gibbs, Heavyside and Wilson, who were following in the venerable footsteps of Euclied, Descartes and Newton.

The Frobenius test

A second test is the division test. If the extension to linear algebra claiming to be a quaternion does not define division then it is not in the classical view isomorphic to the classical quaternion.

The next question to ask is if this new entity is closed under division. This can be problematic for matrix representations that claim to be isomorphic to the classical quaternions. Finding a way to start with two alleged classical quaternions and using the operations of that system arrive at a matrix that does not have an inverse then from the strict classical point of view you have proven that since the system is not closed under division, it is not isomorphic to the classical quaternions.

With these tests in mind a student of the classical view of quaternions can avoid the confusion that results from reading modern commentaries that use the word quaternion in a different context from the context meant in the classical texts.

Classical era nomenclature objection

Hamilton objected to calling the square roots of minus one imaginary numbers, saying there is nothing imaginary about them. 19th century writers on the subject of quaternions seldom if ever use the term imaginary. Instead what from a linear algebra point of view is considered the imaginary part of a quaternion was in the quaternion context called the vector of the quaternion.

Expropriation

Quaternion negative nomenclatures often expropriate terms from classical quaternion nomenclature and redefine them to a point that it makes very difficult to understand classical texts. The i,j,k of linear algebra and the so called scalar product or dot product and vector product or cross product are examples of expropriated ideas. In classical quaternion thinking there is only one product.

Hence although Hamilton coined the phrase vector, and pioneered the concept of four dimensional space and time, quaternion negative nomenclature can construct sentences like "the vector of a quaternion is not a real vector". In the context of classical quaternion thinking this is basically gibberish. It reflects a linear algebra exclusive point of view.

Some modern thinking uses the word quaternion to refer to some extension of modern algebra that has little in common with the classical quaternion. This makes it hard not only to distinguish between their new extension of algebra and the classical quaternion, but between the extensions themselves.

Vectors

See also main article Vector of a quaternion

See also section of this article on the classical vector as an element of a quaternion.

Writers in classical quaternion notation used the word vector differently than it was used by the rival field of vector analysis.

In modern terminology a complex number can be understood to mean the sum of two numbers the first being real, and the second being what we today call an imaginary or purely imaginary number. In modern terms a purely imaginary number is one of the square roots of minus one, possibly with a real number coefficient.

The vector part of a quaternion consisted of three what is today called three orthogonal imaginary units possibly with real coefficients.

The use of the word vector, meaning vector in the classical quaternion context began a steep decline around 1900.

Tensors in linear algebra and classical quaternion theory

The term tensor was introduced by Sir William Rowan Hamilton. Although Hamilton used the word in several different contexts and defined the word tensor differently, for writers working in a linear algebra context, it is commonly, thought of as denoting the positive square root of the common norm of a quanternion. This usage is distinct from the wider meaning of tensor in modern mathematics, which grew out of generalising the norm operation to more general multilinear maps.

In multi-linear algebra tensor means a multi-dimensional array of numbers. The tensor of a quaternion is a single number not an array. What is meant by a tensor now is generally what would be written in quaternion notation as multiple acts of tension, using the product of several quaternions to stretch an object in several directions..

The tensor of a quaternion is a tensor of order zero. When a tensor is multiplied with a vector it has the effect of making it longer or shorter but can never change its direction. The tensor of a vector in quaternion theory has a lot in common with the Euclidean norm. The tensor of a quaternion has the same formula as the R4 norm. But the tensor of a quaternion plays a very different role in H than the norm of R4.

Recalling Euler theorem that a in order to rotate an object from any one orientation to any other orientation requires three angles. Hence a general transform of a vector in space would require three quaternions.

$s(r(qBq^{-1})r^{-1})s^{-1}$

Likewise an entity in general can be subjected to acts of tension in three different directions, by a triple operation. That is assuming that B is a vector in this expression.

$(s+r+q)B$

With proper choice of basis versors the Tensors of these three quaternions Ts, Tr, Tq have some degree of correspond to the modern notion principle axis of the modern stress tensor.Early works on vector analysis called a conglomeration of the somewhat quaternion like entity called a dyad a 'right tensor', and describe a different notion of breaking an operation down into a tensor and a versor. However the versor is closer to a dyadic implementation of a three by three matrix as is stated in the source.

A general transformation of four space would required four quaternions. Beware that unlike Einstein notation, quaternion notation does define powers of quaternions.

$g(s(r(qBq^{-1})r^{-1})s^{-1})g^{-1}$

Since a quaternion always consists of one time dimension and three space dimensions the notion of using subscripts and superscripts to denote covariance and contravariance is somewhat redundant.

Quaternion exclusive point of view

In the classical period some thinkers were guilty of taking a quaternion exclusive point of view.

For example, in the classical quaternion notation system there is only one kind of multiplication or on other words only one kind of vector product.

In the latter part of the classical period advocates of an exclusive quaternion view took advantage of this and used exclusion tactics in reverse. When Gibbs first attempted to expropriate and redefine the word vector, and the symbols i,j and k from the classical quaternion system Tait called the new system hermaphraditical to express his quaternion positive algebra negative bias, in reference to the fact that in Tait's view the new entities apparently had two sets of reproductive organs and belonged in a freak show. Tate was intentionally using a Quaternion Positive, Algebra negative nomenclature by assigning a word with at the time a very negative connotation to Gibbs concept of a vector.

Tait was fond of using the term real distances, for numbers with the square root of minus one. His motive was to use nomenclature that gave his system more appeal.

An example of a quaternion positive nomenclature would be to call the versor of a quaternion written Uq an uncomplicated number.

Frobenius theorem

The 1877 Frobenius theorem is very helpful.

From an algebra context proved that real numbers, complex numbers and quaternions were 'the only possible division rings over the reals'.

From a classical quaternion point of view it draws a clear dividing line between quaternion math and linear algebra by proving what is possible and impossible.

In quaternion math there are three possible closed spaces. A space that exists of just one dimension time or in the context of algebra $R^{1}$. A space that consists of time dimension and one distance like dimension $C1$. A space that consists of three distance like dimensions and one space like dimension. All these spaces are subsets of a quaternion.

Hamilton states early on that any reasonable algebra should include the operations of addition, subtraction, multiplication and division. Hamilton died in 1865, before Frobenius theorem.

Impossible yet useful

A quaternion positive algebra positive point of view sees both algebra and quaternions as useful. Few if any took this view in the 19 century.

An important variant on this last point of view is the quaternion positive algebra positive but impossible view. This view did not exist in the 19th century, but it goes like this:

Just because a space is physically impossible and does not exist, does not imply that it is not useful. In order to solve a system of linear equations in five unknowns the notion of an impossible space consisting of five time dimensions is very useful indeed. Just because Newtonian mechanics is physically impossible because it does not work if you go to fast does not mean that it is not very useful.

Nobody in the 19th century had ever seen anything solid go fast. When in the 19th century Frobenius proved that Euclidean space was impossible, and when in the 19th century Maxwell used the math that Hamilton invented and figured out that light always traveled at the speed of light, which also proved that Euclidean space was impossible, the meaning of these things was incomprehensible.

To the 19 century mind, Euclidean space and the Newtonian physics based on it were both useful and possible. More than possible 19 century thinkers thought Euclidean space was something real. They even called it real space. Real space is an idea so useful that the idea it was also impossible was inconceivable to the 19th century mind.