# Eigenvalues of an unitary operator

## Homework Statement

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?

## Homework Equations

U(U+)=(U+)U=I [where U+ is U dagger and I is the identity operator]

## The Attempt at a Solution

Assume eigenvalues exist
U(a)=x(a) => (U+)U(a)=(U+)x(a) => (a)=(U+)x(a)??

mjsd
Homework Helper
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 $$e^{i\theta}$$" means that magnitude of complex e-vals are 1

dextercioby
Homework Helper
HINT: U unitary means U isometry. Assume the spectral equation

$$U\psi =a\psi$$ (1)

has solutions in a Hilbert space $\mathcal{H}$.

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

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

Dick