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''Hermitian adjoint matrix, of a given (rectangular or square) matrix $A = \left\Vert{a_{ik}}\right\Vert$ over the field $\mathbb{C}$ of complex numbers''

The matrix $A^*$ whose entries $a^*_{ik}$ are the complex conjugates of the entries $a_{ki}$ of $A$, i.e. $a^*_{ik} = \bar a_{ki}$. Thus, the adjoint matrix coincides with its complex-conjugate transpose: $A^* = \overline{(A')}$ where $\bar{\phantom{a}}$ denotes complex conjugation and the $'$ denotes transposition.

Properties of adjoint matrices are:
$$
(A+B)^* = A^* + B^*\,,\ \ \ (\lambda A)^* = \bar\lambda A^*
$$
$$
(AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = A \ .
$$
Adjoint matrices correspond to adjoint linear transformations of unitary spaces with respect to orthonormal bases.

For references, see [[Matrix]].

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