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Eigenvalues/vectors of Hermitian and corresponding unitary

  1. Jan 22, 2015 #1
    Given that any Hermitian matrix M can be transformed into a unitary matrix K = UMU, for some unitary U, where U is the adjoint of U, what is the relationship (if any) between the eigenvectors and eigenvalues (if any) of the Hermitian matrix M and the eigenvectors and eigenvalues (if any) of the corresponding unitary matrix K?
     
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  3. Jan 22, 2015 #2

    HallsofIvy

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    The unitary matrix U, and so its adjoint, has determinant 1 so [itex]det(K- \lambda)= det(M-\lambda)[/itex]. That shows that they have the same eigenvalues.
     
  4. Jan 22, 2015 #3
    Super. S, does that also show that they then have the same eigenvectors?
     
  5. Jan 22, 2015 #4

    HallsofIvy

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    That depends upon what you mean by "the same". We can think of U as a "change of basis" operator. In that sense, eigenvectors of K are the eigenvectors of A written in a different basis. If you are thinking of the vectors as just "list of numbers", then, no, eigenvectors of K will have different numbers because you are writing the vector in a different basis.
     
  6. Jan 22, 2015 #5
    thanks, HallsofIvy. Looking at it that way is very enlightening. That was a great help. :) Understanding is seeping in....
     
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