Why is the conjugate of an eigenpair also an eigenpair

Question

This is an exercise on the book fundamentals of matrix computations 1st. edition.

It asks to show that for $A \in R^{nxn}$, if $(\lambda, u)$ eigenpair, then $(\overline{\lambda}, \overline{u})$ is also an eigenpair.

Answer 1

$A \in R^{nxn}$ and $(\lambda, u)$ is eigenpair

$$ Au = \lambda A$$ $$\overline{Au}=\overline{\lambda A}$$ $$A\overline u=\overline\lambda A\tag{$\because A \in R^{nxn}$}$$

$\therefore (\overline\lambda, \overline u)$ is also eigenpair.