Eigenvalues/vectors of Hermitian and corresponding unitary

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  • #1
nomadreid
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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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  • #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.
 
  • #3
nomadreid
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Super. S, does that also show that they then have the same eigenvectors?
 
  • #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.
 
  • #5
nomadreid
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thanks, HallsofIvy. Looking at it that way is very enlightening. That was a great help. :) Understanding is seeping in....
 

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