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Similar Diagonal Matrices

  1. Nov 6, 2012 #1

    jgens

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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 [itex]X[/itex] is an [itex]n \times n[/itex] diagonal matrix with [itex]n[/itex] distinct non-zero eigenvalues, then exactly which [itex]n \times n[/itex] matrices [itex]A[/itex] satisfy the following equality [itex]AXA^{-1} = X[/itex]? 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.
     
    Last edited: Nov 7, 2012
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  3. Nov 7, 2012 #2

    HallsofIvy

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    Those whose eigenvalues are the numbers on the diagonal of the original matrix.
     
  4. Nov 8, 2012 #3
    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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