Question

Show that every eigenvalue of a unitary matrix is of unit modulus.

Answer 1

Let $U$ be unitary matrix, $\lambda$ be eigenvalue with corresponding eigenvector $x$.

By the definition of unitary operator we have $(x,x)=(Ux,Ux)$ .

But $Ux=\lambda x$ , so $(x,x)=(\lambda x,\lambda x)=\lambda\overline{\lambda}(x,x)=|\lambda|^2(x,x)$ .

Since $(x,x)\neq 0$ , we have $|\lambda|^2=1$ , that is $|\lambda|=1$ .