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Math Help - what's wrong with this argument?

  1. #1
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    what's wrong with this argument?

    So i need help finding what is wrong with this argument:

    Notice A= (A-A*)/2+(A+A*)/2

    and both (A-A*)/2 and (A+A*)/2 are diagonalizable (using the complex spectral theorem)

    thus in some basis A will be the sum of two diagonal matrices and hence diagonal.

    Therefore A is diagonalizable.

    Is the thing that is wrong with this argument the fact that (A-A*)/2+(A+A*)/2 would both need to be diagonalized using the SAME basis in order for their sum to make a diagonal matrix representative of A?

    (A*= the conjugate transpose of A)
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  2. #2
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    You have answered your own question. The basis that diagonalises (AA*)/2 need not be the same as the basis that diagonalises (A+A*)/2. So there is no reason why there should be a single basis that diagonalises both.
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