Is There a Connection Between Conjugation and Change of Basis?

[knowwhatyoudontknow]

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 Ad_g(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?

 

[pasmith]

A and B are also similar if there exists an invertible $S$ such that $A = SBS^{-1}$.