What Matrices Commute with a Diagonal Matrix with Distinct Eigenvalues?

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The discussion centers on the conditions under which an n × n matrix A commutes with a diagonal matrix X that has n distinct non-zero eigenvalues, specifically under the equation AXA^{-1} = X. The conclusion reached is that the matrices A that satisfy this condition are those whose eigenvalues correspond to the diagonal elements of X, and they must share the same invariant subspaces. Furthermore, the eigenvalues do not need to be identical but must be simultaneously diagonalizable.

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As part of a larger problem involving classifying intertwining operators of two group representations, I came across the following question: If X is an n \times n diagonal matrix with n distinct non-zero eigenvalues, then exactly which n \times n matrices A satisfy the following equality AXA^{-1} = X? Does anyone know the answer to this question?

Edit: Nevermind. I found a better way of doing the problem that avoids this sort of argument.
 
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Those whose eigenvalues are the numbers on the diagonal of the original matrix.
 
Is that true? I believe it is the set of operators with the same invariant subspaces. The eigenvalues don't have to be the same, they just have to be simultaneously diagonalizable.
 

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