What Matrices Commute with a Diagonal Matrix with Distinct Eigenvalues?

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The discussion centers on identifying n x n matrices A that commute with a diagonal matrix X having distinct non-zero eigenvalues, specifically under the condition AXA^{-1} = X. The original poster initially sought a solution but later found an alternative approach that bypasses the need for this specific argument. It is suggested that the matrices A must have eigenvalues that correspond to those on the diagonal of X, and they should share the same invariant subspaces. The key takeaway is that the eigenvalues of A do not need to match those of X, as long as A and X are simultaneously diagonalizable. This highlights a deeper understanding of the relationship between the matrices involved.
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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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