Similar Diagonal Matrices

jgens

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.

HallsofIvy

Those whose eigenvalues are the numbers on the diagonal of the original matrix.

Vargo

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.