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Question about diagonalizable matrices

  1. Aug 15, 2013 #1
    Suppose that ##A## is a diagonalizable ## n \times n ## matrix. Then it is similar to a diagonal matrix ##B##. My question is, is ##B## the only diagonal matrix to which ##A## is similar?

    I have thought about this, but am unsure if my answer is correct. My claim is that ##B## is the only diagonal matrix to which ##A## is similar, up to a rearrangement (or permutation) of the eigenvectors of ##A## in an ordered eigenbasis for ##ℝ^{n}##. Thus, if you rearrange the vectors in your eigenbasis, you will obtain a different diagonal matrix for ##[L_{A}]_{β}## but this will be merely a rearrangement of the diagonal entries of B.

    Is this true? If so, how might I go about proving this conjecture of mine?

    BiP
     
  2. jcsd
  3. Aug 15, 2013 #2

    micromass

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    Yes, it's true.

    A hint for the proof: the diagonal entries are the eigenvalues of ##A##.
     
  4. Aug 15, 2013 #3

    CompuChip

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    If ##A = X^{-1}BX = Y^{-1}B'Y## then ##B = (Y X^{-1})^{-1} B' (Y X^{-1})##.
    So if A is similar to two diagonal matrices, then these matrices are similar to each other.
    I guess a starting point would be to check which similarity transformations keep diagonal matrices diagonal, and see what you can say about X and Y that way.
     
  5. Aug 15, 2013 #4

    HallsofIvy

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    Not quite. A matrix can be similar to two different diagonal matrices if they have the same numbers on the diagonal in different orders.
     
  6. Aug 17, 2013 #5

    micromass

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    That's exactly what the OP said:

     
  7. Aug 20, 2013 #6

    mathwonk

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    you might look at the characteristic polynomial.
     
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