Why E's representation marix is unit matrix

[xiaoxiaoyu]

In group representation theory, identity element's representation matrix is unit matrix. But why? Could you give me an exact explanation? Thany you.

 

[Hootenanny]

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.

 

[xiaoxiaoyu]

I know this, but I don't know how to prove that exactly

 

[HallsofIvy]

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= ?