Proof of Hermitian adjoint

Question

How to prove that for any square complex matrix the following equation is satisfied? $$ \langle \textbf{x}, \textbf{Ay} \rangle = \langle \textbf{A}^H\textbf{x}, \textbf{y} \rangle $$

Answer 1

One approach is to simply write out the full sum. $$ \begin{align} \left\langle x, Ay \right \rangle &= \sum_{j=1}^n x_j \overline{[Ay]}_j = \sum_{j=1}^n x_j \overline{\sum_{k=1}^n A_{jk} y_k} \\ & = \sum_{j=1}^n \sum_{k=1}^n x_j \overline{A_{jk}y_k} = \sum_{k=1}^n \sum_{j=1}^n \overline{ A_{jk}} x_j \bar y_k \\ & = \sum_{k=1}^n \left( \sum_{j=1}^n \overline{A_{jk}} x_j\right) \bar y_k = \sum_{k=1}^n [A^H x]_k \bar y_k = \langle A^H x, y \rangle. \end{align} $$