Composition of elements of SO(4)

  • Context: High School 
  • Thread starter Thread starter jk22
  • Start date Start date
  • Tags Tags
    Composition Elements
Click For Summary
SUMMARY

The discussion focuses on the composition of elements in the special orthogonal group SO(4) through Givens' rotations. It establishes that Givens' rotations do not commute, which necessitates careful consideration of their order when combining them. The conversation emphasizes that every element of SO(4) can be expressed as a product of pairs of unit quaternions, with each quaternion corresponding to a specific SO(4) rotation matrix. This relationship is crucial for understanding the structure and representation of SO(4) elements.

PREREQUISITES
  • Understanding of Givens' rotations
  • Familiarity with quaternion mathematics
  • Knowledge of SO(4) group properties
  • Basic linear algebra concepts related to rotation matrices
NEXT STEPS
  • Research the mathematical properties of Givens' rotations in detail
  • Study the representation of SO(4) using unit quaternions
  • Explore the relationship between SO(4) and SU(2) matrices
  • Learn about the implications of non-commutativity in rotation groups
USEFUL FOR

This discussion is beneficial for mathematicians, physicists, and computer scientists interested in rotation groups, quaternion algebra, and their applications in 3D graphics and robotics.

jk22
Messages
732
Reaction score
25
It's a beginner's question : if I consider compositions of Givens' rotations, how many should I combine to obtain the whole group, since those don't commute and hence their order is important ? Is it 6 ?
 
Physics news on Phys.org
Can you be a bit more precise? What are "Givens' rotations"? Do you want to know whether every element of ##SO(4)## can be written as a product of certain elements and of certain length? And most of all: why?
 
There exists an expression of SO(4) elements in terms of pairs of unit quaternions. Each quaternion has a SO(4) rotation matrix associated with it, and the two quaternions' rotation matrices are multiplied together. Those matrices are linear functions of their quaternions, much like SU(2) matrices.

Was that what jk22 was asking about?
 

Similar threads

Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
1K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
988
MHB Question on subgroup and order of the elements
  • · Replies 7 ·
Replies
7
Views
2K
MHB Unlock Role of Correspondence Thm for Groups in Analysing Composition Series
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
962
  • Poll Poll
Why do we spend so little time learning grammar in college?
  • · Replies 137 ·
5
Replies
137
Views
5K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
634
Graduate How to find the number of elements with a particular order in a group?
  • · Replies 1 ·
Replies
1
Views
5K
High School Array Representation Of A General Tensor Question
  • · Replies 2 ·
Replies
2
Views
2K
MHB What Is the Upper Bound of Groups of Order in Finite Group Theory?
  • · Replies 2 ·
Replies
2
Views
2K