Inner product in $\mathbb{C}^n$

I received a question concerning Chapter 3 of the lecture notes where an inner product is used in $\mathbb{C}^n$ (instead of $\mathbb{R}^n$), when doing computations with complex eigenvectors. I thought that you had seen this before, but in case you haven't, please note that the standard inner product in ($\mathbb{C}^n$): $$\langle \textbf{x},\textbf{y}\rangle=\sum_ix_i\cdot\overline{y}_i$$