For $A\in\mathbb C^{m\times n}$, $\langle Ax,y\rangle=\langle x,A^*y\rangle$. Indeed, $(Ax)^*y=x^*A^*y$. Likewise $\langle x,Ay\rangle=\langle A^*x,y\rangle$ when dimensions match. Over the reals, replace adjoint by transpose.
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