# Eigenvalues/vectors of Hermitian and corresponding unitary

Gold Member
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?

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
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
Gold Member
Super. S, does that also show that they then have the same eigenvectors?

HallsofIvy
Science Advisor
Homework Helper
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
Gold Member
thanks, HallsofIvy. Looking at it that way is very enlightening. That was a great help. :) Understanding is seeping in....