Show that the eigenvalues of a unitary matrix have modulus $1$.
I know that a unitary matrix can be defined as a square complex matrix $A$, such that
where $A^*$ is the conjugate transpose of $A$, and $I$ is the identity matrix. Furthermore, for a square matrix $A$, the eigenvalue equation is expressed by $$Av=\lambda v$$
If I use the relationship $u v=v^*u$ and take the conjugate transpose of this equation then $$v^*A^*=\lambda^*v^*$$
But now I got stuck. Can someone help?