##### Actions

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.