A x = \lambda x \ \Rightarrow \ (A - \lambda I) x = 0 \ \Rightarrow \ \langle (A - \lambda I) x, (A - \lambda I) x \rangle = 0 \\
 \Rightarrow \ \langle x, (A - \lambda I)^{\ast} (A - \lambda I) x \rangle = 0 \ \Rightarrow \ \langle x, (A^{\ast} - \bar{\lambda} I) (A - \lambda I) x \rangle = 0 \\
 \Rightarrow \ \langle x, (A^{\ast} A - \lambda A^* - \bar{\lambda} A + \lambda \bar{\lambda} I) x \rangle = 0 \\
 \Rightarrow \ \langle x, (A A^{\ast} - \bar{\lambda} A - \lambda A^{\ast} + \lambda \bar{\lambda} I) x \rangle = 0 \\
 \Rightarrow \ \langle x, (A - \lambda I) (A^{\ast} - \bar{\lambda} I) x \rangle = 0 \ \Rightarrow \ \langle x, (A^{\ast} - \bar{\lambda} I)^{\ast} (A^{\ast} - \bar{\lambda} I) x \rangle = 0 \\
 \Rightarrow \ \langle (A^{\ast} - \bar{\lambda} I) x, (A^{\ast} - \bar{\lambda} I) x \rangle = 0 \ \Rightarrow \ (A^{\ast} - \bar{\lambda} I) x = 0 \\
 \Rightarrow \ A^{\ast} x = \bar{\lambda} x