The combination of transposing and conjugating a matrix will be important in subsequent sections, such as Subsection MINM.UM and Section OD. We make a key definition here and prove some basic results in the same spirit as those above.
If $A$ is a matrix, then its adjoint is $\adjoint{A}=\transpose{\left(\conjugate{A}\right)}$.
You will see the adjoint written elsewhere variously as $A^H$, $A^\ast$ or $A^\dagger$. Notice that Theorem MCT says it does not really matter if we conjugate and then transpose, or transpose and then conjugate.
Suppose $A$ and $B$ are matrices of the same size. Then $\adjoint{\left(A+B\right)}=\adjoint{A}+\adjoint{B}$.
Suppose $\alpha\in\complexes$ is a scalar and $A$ is a matrix. Then $\adjoint{\left(\alpha A\right)}=\conjugate{\alpha}\adjoint{A}$.
Suppose that $A$ is a matrix. Then $\adjoint{\left(\adjoint{A}\right)}=A$.
Take note of how the theorems in this section, while simple, build on earlier theorems and definitions and never descend to the level of entry-by-entry proofs based on Definition ME. In other words, the equal signs that appear in the previous proofs are equalities of matrices, not scalars (which is the opposite of a proof like that of Theorem TMA).