Eigenvalues of an unitary operator

jnazor

1. The problem statement, all variables and given/known data
A unitary operator U has the property
U(U+)=(U+)U=I [where U+ is U dagger and I is the identity operator]

Prove that the eigenvalues of a unitary operator are of the form e^i(a) with a being real.

NB: I haven't been taught dirac notation yet. Is there a way i can do this without it?



2. Relevant equations
U(U+)=(U+)U=I [where U+ is U dagger and I is the identity operator]



3. The attempt at a solution
Assume eigenvalues exist
U(a)=x(a) => (U+)U(a)=(U+)x(a) => (a)=(U+)x(a)??

mjsd

note that you don't need to understand Dirac notation, all you need to know is some basic linear algebra in finite dimensional space. hint: "of the form [tex]e^{i\theta}[/tex]" means that magnitude of complex e-vals are 1

dextercioby

HINT: U unitary means U isometry. Assume the spectral equation

[tex] U\psi =a\psi [/tex] (1)

has solutions in a Hilbert space [itex] \mathcal{H} [/itex].

Then use (1), the assumption regarding the space of solutions and the isometry condition to get the desired result.

jnazor

Sorry i've never heard of isometry or the name spectral equation. I just know it as the eigenvalue equation.

Dick

Isometry means <x,y>=<Ux,Uy>. Why is this true for U unitary? Once you believe it's true set y=x and x to be an eigenvector of U. What do you conclude?