Why E's representation marix is unit matrix

Click For Summary
In group representation theory, the identity element's representation matrix is the unit matrix because it must satisfy the property that multiplying any matrix A by the identity matrix E results in A. This means that for the identity element e, represented by matrix E, the equations AE = A and EA = A hold true for all group members represented by matrix A. The identity matrix is the only matrix that fulfills these conditions, confirming that E must be the unit matrix. The discussion seeks a formal proof of this concept, emphasizing the foundational role of the identity element in group representations. Understanding this relationship is crucial for deeper insights into the structure of group representations.
xiaoxiaoyu
Messages
5
Reaction score
0
In group representation theory, identity element's representation matrix is unit matrix. But why? Could you give me an exact explanation? Thany you.
 
Mathematics news on Phys.org
Welcome to Physics Forums.

Group representation theory essentially boils down to using matrices to represent group elements. In other words, a group is described as a set of linear transformations on vector spaces. In this respect it is obvious that the identity element will be the identity matrix.
 
I know this, but I don't know how to prove that exactly
 
The group identity has the property that ae= a and ea= a for any group member a. Suppose e is "represented" by the matrix E and a by the matrix A. Then you must have AE= A and EA= A, for any matrix, A, in the group representation. Therefore E= ?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Undergrad Need help with tensors and group theory
  • · Replies 6 ·
Replies
6
Views
2K
What is a Group Representation and How Does it Act on a Vector Space?
  • · Replies 1 ·
Replies
1
Views
2K
MHB What is the form of that matrix?
  • · Replies 8 ·
Replies
8
Views
1K
What combination of generators can produce a particular SU(2) matrix?
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Are there two different adjoint representations used in the SM?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad How is unit norm axis rotation represented and derived in R3?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate In what representation do Dirac adjoint spinors lie?
  • · Replies 1 ·
Replies
1
Views
4K
Undergrad Fundamental representation and adjoint representation
  • · Replies 27 ·
Replies
27
Views
3K
Graduate Similarity transformation changing the determinant to 1
  • · Replies 20 ·
Replies
20
Views
3K
Graduate Adjoint representation and spinor field valued in the Lie algebra
  • · Replies 1 ·
Replies
1
Views
1K