Eigenvalues/vectors of Hermitian and corresponding unitary

[nomadreid]

Given that any Hermitian matrix M can be transformed into a unitary matrix K = U^†MU, 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?

 

[HallsofIvy]

The unitary matrix U, and so its adjoint, has determinant 1 so $det(K- \lambda)= det(M-\lambda)$. That shows that they have the same eigenvalues.

 

[nomadreid]

Super. S, does that also show that they then have the same eigenvectors?

 

[HallsofIvy]

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.

 

[nomadreid]

thanks, HallsofIvy. Looking at it that way is very enlightening. That was a great help. :) Understanding is seeping in...