Is There a Connection Between Conjugation and Change of Basis?

In summary, the article explores the relationship between conjugation in linear algebra and the concept of change of basis. It explains how conjugation can be viewed as a transformation that preserves certain properties of linear operators, while a change of basis alters the representation of these operators in a different coordinate system. The connection highlights the underlying structure of vector spaces and linear mappings, illustrating how both concepts are essential for understanding linear transformations and their properties across different bases.
  • #1
knowwhatyoudontknow
30
5
TL;DR Summary
Can the adjoint representation of a Lie group be regarded as a change of basis?
For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix.

For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself.

The expressions have similar form except for the order of the inverses. Is there there any connection between the two or are they entirely different concepts?
 
Last edited:
Physics news on Phys.org
  • #2
A and B are also similar if there exists an invertible [itex]S[/itex] such that [itex]A = SBS^{-1}[/itex].
 

Similar threads

Replies
3
Views
2K
Replies
9
Views
2K
Replies
43
Views
6K
Replies
1
Views
870
Replies
10
Views
3K
Replies
1
Views
2K
Back
Top