Section 3.5 Hermitian Adjoint
The Hermitian adjoint of a matrix is the same as its transpose except that along with switching row and column elements you also complex conjugate all the elements. If all the elements of a matrix are real, its Hermitian adjoint and transpose are the same. In terms of components,
\begin{equation}
\left(A_{ij}\right)^\dagger=A_{ji}^*\text{.}\tag{3.5.1}
\end{equation}
For example, if
\begin{equation}
A = \begin{pmatrix}
1\\
i\\
-2i
\end{pmatrix}\tag{3.5.2}
\end{equation}
then
\begin{equation}
A^\dagger = \begin{pmatrix}
1\amp -i\amp 2i
\end{pmatrix}\text{.}\tag{3.5.3}
\end{equation}
A matrix is called Hermitian if it is equal to its adjoint, \(A = A^\dagger\text{.}\)
Checkpoint 3.4. Try it yourself: Hermitian Adjoint.
Compute \(B^\dagger\) if
\begin{equation}
B=\begin{pmatrix}
1\amp i\\
-5i\amp i
\end{pmatrix}\text{.}\tag{3.5.4}
\end{equation}
Solution.
\begin{equation}
B^\dagger = (B^T)^*
= \begin{pmatrix}1\amp -5i\\i\amp i\end{pmatrix}^*
= \begin{pmatrix}1\amp 5i\\-i\amp -i\end{pmatrix}\text{.}\tag{3.5.5}
\end{equation}
