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 (\(\mathbb{C}^n\)): $$\langle \textbf{x},\textbf{y}\rangle=\sum_ix_i\cdot\overline{y}_i$$