# Eigenvalues/vectors of Hermitian and corresponding unitary

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## Main Question or Discussion Point

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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HallsofIvy
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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.

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

HallsofIvy