For a complex matrix, the adjoint is $A^*=\overline A^{\,T}$, with entries $(A^*)_{ij}=\overline{a_{ji}}$. It satisfies $\langle Ax,y\rangle=\langle x,A^*y\rangle$ for the standard inner product $\langle x,y\rangle=x^*y$. Do not confuse adjoint with the classical adjugate used in inverse formulas.
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