I'm trying to prove that for a certain matrix $A$, and its conjugate transpose $A^*$, we have $⟨Ax,y⟩=⟨x,A^*y⟩$, where $⟨⟩$ represent the inner product. So here it's simply the dot product in $R^n$.
Does anyone have an idea of how to prove this?
Thank you!
Best Answer
The inner product (dot product on $\mathbb{R}^n$ or $\mathbb{C}^n$) is defined as $$ \langle x,y \rangle = x^* y = \sum_{k=0}^{n-1} \overline{x_k} y_k $$ Some definitions switch them around ($x^\intercal \overline{y}$) but they are essentially the same. $$ \langle Ax,y \rangle = (Ax)^* y = x^* A^* y = \langle x, A^* y \rangle $$ Since you are using $\mathbb{R}^n$, the conjugation can be dropped without any problem.