# Eigenvalues/vectors of Hermitian and corresponding unitary

1. Jan 22, 2015

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?

2. Jan 22, 2015

### 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.

3. Jan 22, 2015

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

4. Jan 22, 2015

### 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.

5. Jan 22, 2015