Undergrad Is There a Connection Between Conjugation and Change of Basis?

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The discussion explores the relationship between conjugation in linear transformations and the adjoint representation in Lie groups. It highlights that two matrices A and B are similar if A = S^-1BS, where S is the change of basis matrix. Additionally, the adjoint representation is defined as Adg(b) = gbg^-1, indicating a self-action of the group. The similarity in the structure of these expressions raises questions about potential connections between the concepts. Ultimately, the conversation seeks to clarify whether these mathematical ideas are related or distinct.
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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?
 
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A and B are also similar if there exists an invertible S such that A = SBS^{-1}.
 
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