How can we show that the eigenvalues of a unitary operator are complex numbers of unit norm. Thank you?

Answer 1

U*U=UU*=I def of unitary operator
UX=eX where e and X are eigenvalue and vector, resp.
multiplying the second by X*U*
X*U*UX=X*U*eX
so from first eq and conjugate of second
X*X=e*X*eX
X*X=e*eX*X
X*X is a scalar so
1=e*e
QED