Conjugation vs Change of Basis

  • #1
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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?
 
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  • #2
A and B are also similar if there exists an invertible [itex]S[/itex] such that [itex]A = SBS^{-1}[/itex].
 

What is conjugation in the context of linear algebra?

Conjugation in linear algebra refers to the transformation of a matrix \( A \) into another matrix \( B \) through the relation \( B = P^{-1}AP \), where \( P \) is an invertible matrix. This process essentially represents a change in the matrix representation of a linear transformation under a change of basis, keeping the transformation itself unchanged but altering how it is described with respect to different coordinate systems.

What is a change of basis?

A change of basis in linear algebra involves switching from one basis to another for a vector space. This change affects how vectors and linear transformations are represented. The change of basis is facilitated by a transformation matrix \( P \), which is used to convert coordinates of vectors and matrices representing linear transformations from the old basis to the new basis.

How does conjugation relate to change of basis?

Conjugation is a specific application of the change of basis concept. When you conjugate a matrix \( A \) with a matrix \( P \) to get \( B = P^{-1}AP \), you are effectively representing the same linear transformation in a different basis. The matrix \( P \) serves as the change of basis matrix that transitions from the basis corresponding to \( B \) back to the basis corresponding to \( A \) and vice versa.

Why is conjugation important in linear algebra?

Conjugation is crucial because it allows us to explore properties of linear transformations that are invariant under basis change, such as the trace, determinant, and eigenvalues of a matrix. It also helps in simplifying matrices into more manageable forms like diagonal or Jordan canonical form, making them easier to analyze and work with in various applications such as solving differential equations and optimizing algorithms.

Can any matrix be used for conjugation and change of basis?

Not every matrix can be used for conjugation and change of basis. Only invertible matrices (non-singular matrices) can be used as the matrix \( P \) in the conjugation process \( B = P^{-1}AP \). This is because the operation requires \( P^{-1} \) (the inverse of \( P \)) to exist. If \( P \) is not invertible, its inverse does not exist, and thus the conjugation cannot be performed. Choosing an appropriate \( P \) that is invertible and aligns well with the structure of \( A \) is key to successful conjugation and effective change of basis.

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