Maths - Quaternions- Datasheet

For more information about quaternions see this page. For infomation about how this page was derived see here.

Algebra Laws

Complex Numbers over the real numbers are a 'field' they have the following properties:

  addition multipication
unit element 0 1
commutative yes no
associative yes yes
distributive over addition - yes
inverse exists yes yes

Multiplicative Group

If we ignore addition and treat complex numbers as a group then we have:

Cayley Table

  1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
i i -i -1 1 k -k -j j
-i -i i 1 -1 -k k j -j
j j -j -k k -1 1 i -i
-j -j j k -k 1 -1 -i i
k k -k j -j -i i -1 1
-k -k k -j j i -i 1 -1

Cayley Graph

quaternion cayley graph

For more information about Cayley graph see this page.

Cyclic Notation

As can be seen on the Cayley graph above each generator has two 4-element cycles :

<(1 4 2 3)(5 8 6 7),(1 6 2 5)(3 7 4 8)

For more information about cyclic notation see this page.

Group Presentation

There are two generators.

<i,j | i² = j², j -1i j = i -1>

where: k=ij

For more information about group presentation see this page.

Group Representation

These are the root of -1 the identity matix. See this page for information about taking roots of a matrix. There are two such generators:

[i] =
0 -1
1 0
[j] =
i 0
0 -i

k can be generated from i*j:

[k] =[i][j]=
0 i
i 0

For more information about group representation see this page.

Related datasheets


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.

Terminology and Notation

Specific to this page here:

 

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